Yes, the recipe can be baked...
For cheese, I use what I have on hand...
cheddar, american, monterey jack, gouda, swiss... whatever, as long as it is a soft cheese...
if using swiss or gouda, since they are "stringier" I like to mix them with other cheeses, like cheddar or american...
also don't let the sauce boil, just a simmer, or even better use a double boiler when you add the cheese (then it will be really creamy!) or the cheeses will "break"...