Do you have any experience with music in general? I mean, do you play any instruments and/or have a decent knowledge of chord structure and music theory?
It depends on how proficient you need to be with the program in question. You can learn many of these programs will enough to do a simple, basic arrangement in a week or two. Learning all the ins and outs of the program, learning to get the best possible sound from the synths/samples you're using, learning how to get the highest-quality recordings for any live instruments you may be using, this is an ongoing process which you never really "finish." I've been doing this for years and I still feel like I've barely scraped the surface. But I know enough to make music that I'm happy with.
Again, there isn't really a straightforward answer to this. You can do some pretty spectacular stuff using only free software/samples/synths/effects. However, if you have some money to throw around, you can get higher quality sound much easier in many cases.
As far as how much you'd be spending, the sky's the limit. If you can give us an idea of your budget, we can probably steer you in the right direction, but there isn't really any way for us to say "you need to spend exactly $XXX to be a good remixer." DarkeSword, Protricity, and Unknown use free sample almost exclusively. On the flipside, I've spent upwards of $20,000 on hardware and software specifically for making music -- I know some of the pros around here (like zircon) have probably spent a LOT more than that.
I think they sort of come together. You can't really separate the two, in my opinion. Just practice, practice, practice.
Not that important, in my opinion. I'm sure some people will disagree with me on this point, though. I generally go by ear and stick with whatever sounds good. I often don't bother figuring out the exact chords of the original.
Having a general understanding of the structure of the original is important, but knowing every note played by every instrument in the original track isn't.
Yes. Like anything, formal education is probably helpful, but most of the remixers here (including myself) are self-taught.
I'd guess quite often. But I can't be sure; we generally don't keep track of people who try once and then quit unless they make a complete ass of themselves afterward.
I'll tell you right now: your first shot at remixing will probably suck. You'll think it's awesome, you'll sub it, the judges will tear it apart. Don't get discouraged and don't take it personally; it's like that for everybody.
I know all these vague "it depends" responses probably weren't what you were hoping for -- fortunately/unfortunately, making music is a very individual process, and everybody does it differently.
Best of luck!