how do they determine game size in a game without discretized bet sizes? what assumptions do they use? what assumptions are allowed to be made while considering a nl game to be "solved?"

Why do you think the bet sizes are not discrete in NL? One of the researchers of the University of Alberta invented an algorithm to calculate the size of No Limit games using a recursive procedure described here http://poker.cs.ualberta.ca/publicat...rt-nl-size.pdf

I took his code and ran it for 25bb that's how I got those numbers. However since blinds increase in hypers, the game should be a tad smaller than the number I quote above.

As for the last question, solving full NLHE would mean finding Nash Equilibria in the full game tree which appears to be an intractable problem. AFAIK all the research is being made in abstracted (coarser) game trees which they hope can be translated to strong strategies in the real game. It's a complicated subject but they make the games smaller by merging similar game states and limiting the number of actions (bet sizes) at decision points.

That's not true. The game is bigger because you have to solve it for every possible effective stack size.

you are right.