At first glance, your question looks like a shopping-cart question. But in reality, as of when I write this post, there are only two viable pieces of software for what you're looking for: Ableton Live and Bitwig Studio. (Bitwig Studio is also available for Linux, although it isn't FOSS.)
Ableton does indeed do everything you want it to. It is full-featured enough to easily be controlled by MIDI messages and to send MIDI messages back out to external hardware. The demos you saw only used the mouse because that's the most common and immediately available input device, and Ableton, being a professional-grade piece of software, has too much functionality to show off in a single video. Bitwig Studio is just as powerful.
You asked several questions in your post, so I'll go through them in detail one by one. Before I begin, though, I need to acknowledge an unpleasant reality: Computer music is an expensive hobby. I'm going to recommend a lot of software and a bit of hardware to you. If you buy all of it, you can very easily spend several thousand dollars. Keep in mind that my answer is meant to be a broad overview of everything that's possible with computer music. You do not have to buy everything I mention to be successful. And if you do want to eventually have a very complete setup, you can acquire it piecemeal over time. The biggest thing you need is the first item, a good DAW or VST host. Everything else can be picked up later, and you can also go for a cheap DAW tier or cheap VST host for now and upgrade later.
Is there a computer workstation-based setup that could take MIDI events from my keyboard, and send them back to my synth's sound generator, adding arpeggios and MIDI patterns depending on my inputs?
Yes, absolutely. What you're looking for is either a DAW or a VST host. Again, of the options available on the market, I think that Ableton Live or Bitwig Studio are the most likely to do what you need.
A DAW, or digital audio workstation, is a piece of software designed for creating computer-produced music. Any decent DAW will be extremely full-featured, allowing for playing raw audio files alongside MIDI-controlled instruments, sending MIDI to and from external devices as well as virtual software instruments, and much more.
The reason I recommend Ableton Live and Bitwig Studio is that, more than any other DAW on the market, they are both designed with live performance in mind. Most DAWs have a linear timeline, similar to the editing track in movie editing software, that plays a song from beginning to end. Ableton and Bitwig add an additional mode, called Session View in both pieces of software, that is focused on live performance. This mode lets you organize MIDI clips and sound files into lanes that can then be triggered by clicking on them - or by programming and sending MIDI messages that switch between them. It is very much exactly what you are looking for!
I use Ableton Live and have used it for some live performances. I wrote some music for a Dungeons and Dragons campaign I'm running. By programming Ableton to work with my keyboard's touch pads, I had it set up so that I could very easily change the music in response to what was going on without even having to see Ableton on my computer. Touching the right button on my keyboard queued up sometimes upwards of a dozen MIDI clips at once, kicked them off at a musically sensible time, and automatically worked through musical transitions before looping the main themes until the next message.
You can, of course, also play music on your keyboard and have that come through at the same time as all of the music you queue up in advance. It takes a little bit of work, but you can also configure your keyboard to play multiple instruments at once by splitting key ranges.
All of this functionality is available in Bitwig as well. (In fact, Bitwig was created because Ableton was the only DAW on the market with a live performance mode, and the creators of Bitwig saw an opportunity in setting themselves up as a direct competitor.) Bitwig is also Linux-compatible, so it might be even better for you.
Another option, one that is much less expensive but also less powerful, is to use a VST host. A VST host is a piece of software that lets you connect a MIDI keyboard to your computer to control virtual instruments. The downside is, unlike a full DAW, a VST host does not let you write MIDI clips or program clips to start playing in response to MIDI messages. There are ways around this if you're willing to invest in looper plugins, but they're far more cumbersome to work with than the immediately-available editing environments of Ableton and Bitwig.
There are a lot of VST hosts out there, so unlike with the two DAWs, I don't feel comfortable strongly recommending one to you. You can do some research by Googling "VST host" and seeing what the options are. This blog post also has a very good overview of some of the better options as of when I write this post, but it will likely go out of date in a couple years: https://blog.landr.com/best-vst-host/
Whatever you go with, I strongly recommend trying free trials before committing to one of these options. Music production software is very expensive, and getting a good DAW is one of the biggest purchases you can make in the hobby. Ableton will set you back $100 to $750 depending on which tier you decide to get, and Bitwig costs $100 to $400 but also requires you to pay for yearly major upgrades if you want to keep it completely up to date. VST hosts usually run $50 to $120. Bitwig and Ableton both have extensive free trials that give you access to most of their functionality, so you can see for yourself whether it's comfortable creating and using the setup you want in them before spending your money.
Can I extend such a system with so-called virtual instruments to add more sounds to my setup?
Absolutely! In fact, this is one of my favorite parts of computer music. There is an extremely rich world of virtual instruments out there to explore and play with. And any good DAW or VST host will put them immediately at your fingertips when you sit at your keyboard.
The market for virtual instruments is, frankly, extremely saturated. There are far, far too many companies offering too many options for me to begin to recommend certain options over others. But I'll give a very quick overview of what to expect from the software, some of the biggest names, and how to go about looking for the software you want.
There are four main formats for virtual instruments:
- Stock instruments come bundled with DAWs. Ableton and Bitwig both come with libraries of excellent sounds built-in, and the higher the tier you buy into, the larger the library you have access to from the get-go. Depending on what you want to do, you might find that it's actually cheapest to buy the highest tier available and rely on the stock sounds, which will give you an extensive library from the get-go.
- VST instruments are the lingua franca of virtual instruments. A piece of software in this format will load in very nearly any DAW or VST host you choose.
- Audio Units are an alternative to the VST format that only work on Apple computers. You're unlikely to be able to use Audio Unit-only instruments since you seem to want to switch to Linux or perhaps Windows.
- Synth presets are configurations for virtual instruments that program them for you. You can only use a preset if you also own the VST or stock instrument that it programs. Of particular note are Kontakt and Omnisphere libraries. I'll discuss them more below, but Kontakt and Omnisphere are two of the highest-grade synths on the market and are sophisticated enough that they can make it sound like a human is playing your synthesized instrument. You'll very likely want one or the other if you are planning on playing styles like classical music, musicals, or jazz. But be prepared to spend a lot of money.
Here are some virtual instruments that are especially noteworthy:
- Vital is a free and open-source synthesizer for EDM and pop. It's noteworthy for being one of extremely few FOSS instruments available that is also professional quality. If you want to get started without spending a lot of money and play pop or EDM styles, it is an excellent starting point. The source is available at https://github.com/mtytel/vital.
- Serum is arguably the most highly-regarded synthesizer for EDM as of when this is posted. One excellent upshot of this is that there are hundreds of thousands of presets for the instrument being sold, so if you want a specific flavor of EDM sounds for your performance, you can almost certainly find plenty of preset packs for it.
- Kontakt is arguably the most highly-regarded instrument for replicating acoustic instruments. The best Kontakt libraries can be completely indistinguishable from listening to a live orchestra or musician. Kontakt itself is very expensive, though, and good libraries are also very expensive. Kontakt costs $400, and most individual libraries cost $20 to $150. The highest-end libraries cost several thousand dollars! If you do decide to get Kontakt, I very strongly recommend getting one of the Komplete bundles that go along with it. Native Instruments, the producer of Komplete, makes middle-tier libraries - not the best, but excellent for getting started with a well-rounded library. And while the bundles are quite expensive, the amount of content they give you access to is very generous for the price. If you want to save some money, wait for the 50% off sale Native Instruments offers every year in June.
- Omnisphere is the biggest direct competitor to Kontakt. I'll admit I haven't had experience with Omnisphere myself, so I can't speak to how good it is myself, but I have noticed that it's well regarded. Like Kontakt, Omnisphere is expensive itself, and good libraries for the instrument can oftentimes cost several hundred dollars on top of them.
If you don't want to spend that kind of money, you can look for free instruments instead. Although Vital is the only free instrument I'm aware of that's worth pointing out explicitly, there are countless thousands of free instruments around the internet for you to check out. It's rare to find one with the pristine sound of a high-end instrument, but you can build a perfectly fine library out of free instruments if you're willing to put in the time to look around for them.
In addition to virtual insturments, VSTs also provide virtual effects. These are tools that modify the output of instruments before playing them out the speaker, adding things like delays, reverbs, and mixing/mastering controls. Think of them like virtual guitar pedals. You don't need them to perform live, especially if you're in an environment where you're playing in a band with a reliable sound engineer controlling the deck, but they can also be very useful. If you use Ableton or Bitwig, both come with outstanding libraries of audio effects; you can very plausibly get by just using them.
One audio effect I do want to recommend, though, is Venomode Phrasebox. This effect takes the MIDI you play and processes it before sending it to the virtual instrument. It allows you to create elaborate arpeggios, turn chords into melodic lines and basslines, and much more. There are other arpeggiators out there for cheaper, so it's worth looking around at other options if it looks like something you're interested in. From what I'm aware of, though, I haven't seen a MIDI effect with as much flexibility as Phrasebox anywhere else, and it looks like it might be something you're interested in in particular if your goal is to be able to play as a one-man band with a MIDI keyboard. In addition to queueing up MIDI clips and audio files in Bitwig or Ableton, Phrasebox would let you hold down a chord to control multiple lines simultaneously.
And in the future, can I do the same with only a MIDI controller keyboard and a laptop, when my synth dies?
Absolutely! In fact, I have never had the chance to use a hardware synthesizer. I understand they can be a joy to use, but I have also very much enjoyed working with software synthesizers. Everything I've discussed in my answer is very much possible with just a MIDI controller, a DAW, and a decent library of software instruments.
One last note I want to mention: You might also consider getting a MIDI pad controller, like Native Instruments Maschine or Ableton Push. These controllers consist of a grid of square buttons. They do not have remotely the flexibility for playing lines of music live like a keyboard does, but that's not what they exist for. Instead, if you go with Ableton or Bitwig, you can program each button to trigger one or more MIDI clips or audio files. With a keyboard and a pad controller, you can easily perform your main line on your keyboard while controlling the rest of the song through the pad and the pre-programmed clips you have on your DAW. (Some MIDI keyboards come with some pad buttons on them already, so you might not need a separate pad controller.)
