First of all, "making professional sounding, high-quality productions" takes years of experience and investment in decent quality gear. So, don't be disappointed if you can't "jump right into" it. Unfortunately, as with anything about music, patience is the key here.
I would recommend you to start with reading some books on recording, mixing and mastering; probably in that order. All three are equally important and a good recording engineer is not necessarily a good mixing engineer or a good mastering engineer (and vice versa). They are different domains of expertise.
A good producer needs to know about all these fields, at least enough to know who to hire. Producer's job is to conceptualize the final product from the beginning and direct the song writers, performers and recording, mixing and mastering engineers towards that concept (whether he does or does not perform one/some/all of these functions).
Now, about your specific questions:
Recording acoustic drums
It depends on the genre. For jazz, you can get away with a pair of good overhead mics and a good drummer with good control, possibly complemented with a good bass drum mic. But if you're aiming at processed rock drums for example, you're going to need more than that so you can process the bass drum and the snare drum separately at the very least. Drum mic setups with 10+ mics are not uncommon.
There are cheaper high-quality mics on the market now, but it's still quite an investment. Also, the quality of the recording room is at least as important as the quality of the drum set and the mic set, if not more so. You will also need good mic preamps and a good sound interface.
The only cheaper option I can think of is using a MIDI drum set (or writing MIDI data by hand) and a good software drum sample library. Depending on the genre, they can be totally unacceptable or better than real drums.
Believe it or not, all major recording software (or DAWs -digital audio workstations- as they call them) have more or less the same functionality and the same sound quality. ProTools can come with some external cards that takes the CPU load off of the main computer but with a powerful computer it's not an absolute necessity. The choice is more about workflow habits (and budget!) than sound quality.
You can buy a cheap DAW (like Reaper) and spend the rest of your money on effects and synth plugins and/or external gear. In fact, effects and synth plugins are more important than your DAW.
Crisp snare drum
I don't feel qualified to answer this question but you can definitely achieve good results with most crisp sounding snare drums with proper room, mics, preamps and effects (eq, compressor etc.).