I am far from expert on this issues, but I will try to answer to the best of my knowlege

Is it customary to go to Israel to do it or can it be done in one's home country?
As far as I know there are Yeshivas in other countries as well, but Israel as a religious center of the jewish religion has a lot of them and to my knowledge some of the most respectable ones
Think of it as a form of foriegn exchange students, when people from other countries come for a year of study


Is gambling inconsistent with Jewish religion, according to popular opinion?
Is gambling incompatible with Yeshiva? Again according to Wiki, it's permissible to work during Yeshiva, so if one sees themselves as a poker player by profession, is that OK?
One of the regulars in my home game went through the process of accepting and practicing his religion seriously, when he still used to come I had some discussions with him about it, to my very limited understanding gambling is a very shady area, it is considered an 'unhonorable' way of earning an income as you take money from another without really giving him something in return, so as a profession or any kind of games for serious money it is not allowed, as far as more friendly game for fun for a relatively small amount of money it is less clear and I assume it is pretty much the Rabbi's call