If I tell Sticky to wait, say, in my house, and come back later, will he be dead?
If not, how long will he wait in my house?

He will wait until you come back. The only "Sticky" glitch is that he sets off your mines. So if you've mined your home, he will definitely be dead. I've had him wait up to two weeks in a location before he takes off on his own.

No he'll never leave (as far as I know, I've used him quite extensively and always found him in the spot I left him). Although he does tend to "sneak" around the area you leave him in so he might not be in the exact same spot.

He's a loyal and true companion. He'll never abandon you. :disguise:

I can't seem to heal him.
We got jumped by a sentrybot, so I went to explode the car the bot was next to, and Sticky ran up to him with a sawed off and took a good amount of damage. I tried traveling, waiting and sleeping. He's still very very low on health and suffering a broken leg.

It's a bug. I forgot how to fix it... Sorry :/.

actually I don't think it is a bug. If I remember correctly, Sticky doesn't heal on his own, like most NPCs he requires a stimpak in his inventory.

So trying putting a few stimpaks in his inventory Robotoh and when he gets engaged in a fight next time, he should heal before fighting.

I threw like 5 into his inventory when I noticed he was messed up. I hope that works. Thanks.

Just remember to give him time to heal. Its not going to do much good if he runs up and is shot before he injects the stimpak.

I wouldn't bring Sticky to your house if I were you. Chances are he'd never leave, and you'd be stuck with him telling you stupid stories and eating all of your dandy boy apples.