User blog comment:Hyperborean/"The House of Hades" – official expectations blog/@comment-6266020-20130426213518/@comment-3146930-20130426214821

Tartarus is the place where monsters reform so theoretically if one was killed, it could instantly reform. However, because Gaea brought the Doors of Death to Tartarus (or they were in Tartarus to begin with, which is unlikely), monsters are able to easily pass into the Mortal world which is why they are reforming so fast.

For your second question, yes Percy and Annabeth could die in Tartarus. Even if they are technically in the deepest part of the Underworld, they aren't dead. If they died, their souls would just go to the Judgement Pavilion and they would decide where their soul should go. It is also possible that because nothing is supposed to be able to escape Tartarus, Percy and Annabeth's souls could be trapped there forever. If Percy and Annabeth did die and tried to use the Doors of Death to escape the Underworld, they would still be considered escaped souls and not be truly living anymore.

Also, it does matter if the Doors of Death are open. Even if Thanatos is free, he can't completely stop the flow of monsters and evil souls from leaving the Underworld, as Gaea is more powerful then he is and he can only slow them down. In fact, it may be only a matter of time before he is captured again.