"You look Time Lord."

Actually, the TARDIS doesn't have a human body and it was made clear the body of Idris was dying.

Good point. It wasn't the TARDIS' body because she wasn't native to the body, and calling it hers makes it sound like a house or other possession that could be bought or sold. It's disrespectful to Idris's mortal coil and therefore to Idris, and that was not my intent.

Some may say it was done before: But why couldn't have House and TARDIS change places. Yes, he would have been in Idris' dying body, but it worked for Cassandra.... She was able to accept her death.

I don't think this was an option. TARDIS wasn't released until the body died, period. Sticking House in a dead body seems problematic if even possible (zombie!House, anyone?). The difference between sticking anyone in a about-to-die-within-five-minutes body and killing them outright seems like mere semantics to me: you're still condemning them to death and their death is still on your hands. Yes, one gives a less violent death than the other and a little time to come to terms (if you can do that fully in five minutes or less), but ultimately death is death and killing is killing.