Axolotls can live with just an air-stone. I believe this is how those being used for research are generally kept, could be wrong. But this method of housing requires large proportion water changes daily (depending on size of tank) if there is no filtration. I keep my baby larvae in tubs with an air-stone and they do fine. One set-up uses a sponge filter. But once they are adults they will need more room and better filtration due to the amount of ammonia they produce once more developed.
Your female could definitely go back in with the male after being separated, but then again they will yet again continue to breed. I'm in the same situation. I don't have the space to separate my pair, so I'm trying to down-size/sell off some of the fish from other tanks.
Karipatra, if you're interested I make it to Seattle every now and then and have a few hundred babies. PM if you want.