A - If I'm adiacent to a hostile miniature and attack him using Explosion 1, did I suffer 1 damage?
B - If I push a miniature and there aren't free adiacent spaces, that miniature move in the nearest free space or did not move?
A) Assuming you mean the keyword "Blast" then yes. Attacking an adjacent figure with "Blast X" causes you to take X damage
B) No. Assuming that it isn't massive and you are only getting to push it one space, it cannot be pushed to any of the spaces then.