And they aren't really related, either.

Can I kill house whenever I want? Can I do all of the missions and then take him out?

Also, is the speed boost given to Rex when you choose his brain any good? Is it worth not getting the attack bonus or DT? Not playing on hardcoe so I guess DT doesn't matter as much.

I completed the quest with Lupa's brain then returned Rex to the King, haven't used him as a follower aside from that.And with House, you can if you want to.