The law degree per se isn't going to get you into Belgium, France or any European country other than if you manage to land a job first. (For Belgium, you could try NATO - no visa needed if you land a job there.)
About France - straightforward, perhaps, but not quite as automatic as you put it.
It can be difficult to get a visitor visa if you are of "working age" (because, I suspect, they don't really believe that you won't be tempted to try to supplement your income by working under the table) and have no other plausible tie to France (fiancé, study plans, research project, etc.). Being a day trader isn't considered a stable source of income, and to do so in France, you'd need to hook up with the French tax and cotisation (social insurance) system to pay in your fair share.
After 5 years you can apply for nationality - but the process takes a good year (during which you have to be street legal - i.e. with a proper carte de séjour, which is your residence permit) and prove that you know the language at a B2 level plus are properly "integrated" (which these days tends to mean that you have been paying in your fair share during all or most of the 5 years you've resided in France). It's not automatic, and I know people who have been turned down for nationality, even after 20 years or so of living in France.
Cheers,
Bev