Some cards say they get new abilities at level 2, 3. Do they level up with crafting?

No, they level up during gameplay. Cards in play with levels gain Growth Points equal to the MP cost of friendly units they see you play from your hand. When a card with levels has gained Growth Points at least equal to its printed Growth value, it will spend those to gain a level.

I haven't played with Leveling cards enough yet to notice whether a card will waste excess growth points or store them towards gaining the next level, but this should be enough to get anyone started with using this game mechanic.