How do composers (professional or not) get their works performed by orchestras and soloists?
While we're at it, how and where do composers publish their music and where do orchestras/musicians/conductors/music directors find this new music?
These are great questions. BIG questions, but great nonetheless.
How so works get performed?
There is an old composition adage: first you write for yourself, then you write for your friends, then you write for others. This basically sums it up. If your friends trust you and you do a good job, they'll keep playing your stuff. If you're lucky enough, people who aren't your friends might notice and want to play your stuff too.
Orchestras, because of their size, are always a bit of a luxury in terms of performance. Many orchestras hold composition competitions in which they'll premiere and record a winning work. Other orchestras hold workshops / intensives where they do the same thing.
A lot of composers do what's called "cold-calling" with directors, sending them a score with a recording or a note. Most directors don't like this. ALL directors have a stack of about 2-3ft of scores they've been sent. When they feel nice, they'll pick one out at random and take a look. Sometimes it can take 10 years before they get to your piece, and even then they might not like it.
If a conductor really likes you, they'll come to you for projects. It's much, much easier if they're your friends.
Soloists are a little different. Depending on the level of the soloist, they might be really flattered that you sent them music. After all, it helps you both out. Just like writing music, projects come about all different ways. For example, I was a guest composer at a festival, and a horn soloist liked one of my pieces so much, he commissioned me right there for a piece for the following season. I didn't know him at all then, but now he's a colleague and friend. Again, all this stuff is easier if you're working with your friends.
Also a big topic here. I will scratch the surface like I did with the other question. Some composers work exclusively for publishers. These are mostly the "educational" composers that feed kids' brains. Band. Chorus. Etc. Many more composers are self-publishing now or are published through boutique publishing houses representing a small number of composers (10-30 usually). It is not hard to self-publish, but there's a big trade between how much you own versus how you're marketed.
A final thought related to this subject as I've seen it pop up in other answers and I know my posts can have a long reach:
Networking is bullshit. I'll say it again: Networking is bullshit. You either have friends and people you care about or you don't. Plain and simple. Otherwise, you're just using people, and if you think they can't sense that a mile away, you're silly and they'll never work with you, or worse, they'll work with you once and be done forever.
You'll just come across as being fake and disingenuous. Just find people who dig you and the stuff you do. Find people who you dig and the stuff they do, and make stuff together. Whatever "success" happens from that is just gravy.
I can only answer the first question, as nothing of mine has ever been published in spite of winning a first prize and an honorable mention in a prestigious contest.
The way I get my work performed (and I know a lot of other composers use the same approach) is to make friends with an established conductor or assistant conductor, do him or her a lot of favors connected to the orchestra (NOT favors for the conductor personally), and when he or she eventually gets around to asking about my music, say, "Well I have this piece I wrote that I'd love to hear." The conductor will usually schedule it for a run-through at a rehearsal. If the conductor likes it and the consensus among the musicians is that they wouldn't mind performing it, it gets played at a concert. Other musicians in the geographical area spread the word and some other conductor approaches me, etc., etc. I don't have the kind of reputation that has people from timbuktu coming after me, but it works at a low level.
Same with soloists, most of whom I meet through the conductors. Though one of my soloists is my Voice Teacher!
In other words, Networking. Just like you would to get a job. (Which is what you're doing. Sort of.)
When I was in high school and college, I got a few pieces performed (no money) by writing incidental music (and a few songs and dances) for Little Theater groups in my home town. You might approach an amateur theater group (or even an amateur theatre group if you are in England) about supplying some incidental music. Fortunately the directors of these groups generally seem to give vague directions of what to write so you do get a bit of freedom. The point is to get name recognition. (And you get some experience writing music that someone actually wants.)
Self-publishing your own adaptations of already existing music is an idea, get a person from your local conserve to play it and publish it on youtube, great exposure fore hopeful performer and composer alike, remember to put a link to the sheet music in the description.
Build a name for yourself as an arranger first, working with existing music first is just easier as the audience already exists. In many ways skillful arrangers are rarer than skillful composer, you are less likely going to make living becoming the next Mozart but make a jazz adaptation of Eine Kleine Nachtmusik and get it played on a dog food commercial and you would be surprised just how rich you become.
This is just an easier way to both get your name out there and also grind your teeth until those incisors are so sharp that when the day comes that you compose something you already have someone to play it for and what you play will be of a good quality.
A non-professional's story:
In college I was fortunate enough to have some student compositions performed by the student chamber choir and the flute professor. The choir piece was 2 or 3 takes of sight reading during a rehearsal. That was suggested by the choir director when I presented the score to get his feedback. The flute piece - an independent study assignment - was played at a student concert and my composition professor asked the flute professor to play it for me. That one too was really a sight reading. In both cases the people let me record them with a boombox during rehersal. In I was only a music minor so these professors were being really generous.
I realize in retrospect how lucky I was and how important is was that the music was sight-readable. If/when I try to write again, I will keep this in mind for the sake of some chance of a real performance.