I look at my professors, they are almost always there to do research, day and night, weekday and weekends, even the ones with family. The more leisurely ones are the ones who either got their professorship or published some great papers a while ago. Otherwise, everyone else is very busy.

First of all, thank you for realizing that. I'm serious. Most people seem to believe nowadays that math. professors are the laziest people in the world...

Now back to the question.

A lot depends on what you want to achieve. If you want to stay in the "low middle tier" as a mathematician, you can perfectly well combine mathematics with social life. Moreover, you'll have excellent opportunities to travel, to design your own schedule, etc. You'll still need to work 6-10 hours a day to be worth anything and you are perfectly correct when saying that the outcome is not directly proportional to the effort in mathematics on the daily basis (though, on a very long run, it still is) but it is actually a much better life than that of a factory worker or an office clerk. If you want to move up from there, it'll take more effort. Slightly above the middle tier, you have to "run as fast as you can just to stay where you are". Then your social life may get screwed up quite a bit.

Money has always been a problem and will always be. Short of marrying a rich girl or becoming a single heir of a millionaire uncle, nothing will solve that problem immediately, so prepare for quite a few years of "frugal life". What may help is tutoring, occasional consulting, taking some odd jobs, etc. See what you can do with the skills you already have. Many (now) famous mathematicians survived that way when they were students.

However, without formerly enrolling in school, my work will be mostly unpaid since their funding comes from NASA and other institutions, and I don't have a PhD to qualify me as a visiting scholar to let me access school's supercomputers legally.

This tells me that you are more into the computer science than "math. proper". There is nothing wrong with that. What I said above applies more or less to any "skilled trade".

About job opportunities. Be careful what field you choose. There are two extremes: to choose whatever you like most and bet on your becoming so good that they'll have to notice you no matter what and do things the way you want whether they like it or not. The problem is that you may be not strong enough and then this approach will fail quite dramatically. The other extreme is to follow the current fashions and be in some "hot area". Then you will have to follow the trends whether you like them or not but you are almost guaranteed a decent job. The problem, of course, is that fashions change with time. Most people end up somewhere in between (except the lucky guys for which the two choices coincide).

About student years. Yes, those are most demanding. You've got to put a lot of effort into your training to be able to pass the threshold above which you become a "professional". Prepare to face the fact that "you are nobody until you prove otherwise". You'll, most likely, be treated with respect, given help, etc., but if the aforementioned proof is missing by the end of your student years, your future won't be very bright. I'm not talking about the formalities here: more or less everyone can get a degree. However, to build a lasting reputation is much harder and it is your reputation, not your formal degree, which determines your opportunities after you graduate. There are, of course, technical things like going places, giving talks, letting people know you, etc. but, to be frank, to arrange your meeting with any mathematician of any rank is not too hard and most advisers are good at that. The problem arises when you are there and it is time to show something.

A small comment on what Alexandre said. Math. research is a hobby in only one sense: you aren't paid for it directly. However, in every other respect it is like any other art or craft: as Paganini used to say "If I do not play one day, I hear it. If I do not play two days, experts hear it. And if I do not play three days, the public hears it". Of course, "day" means some longer (but not much longer) unit of time here, but the message is clear: it is not something you can do on an occasional basis when you feel like it (unless you always feel like it, which is the best case scenario as far as the motivation goes).

So, that's more or less what it takes (from my point of view). What are the benefits? Well, the main one is that you become a dual nature creature that is not bound to this world alone, but to explain what I mean by that will take too much time. Let's stick to a decent salary, great work freedom, meeting a lot of interesting people, etc. Of course, you still have an option to turn it all into hell (The best way to achieve that is to quarrel with your colleagues over some trifle issue of this world and the quickest way to achieve the latter is to look for the general fairness and ethics with concentration on what is unfair with respect to you personally. The second best way is to start worrying about your prestige and priority.) But if you avoid these traps, you'll like what you get in the end.