First of all, here's
the thread that shows you how to do a salt bath (btw, you will need to buy aquarium salt for this...and it doesn't cost too much. A few bucks for a container that'll last you a while). I found this thread extremely helpful and I hope you will too.
If the bathroom is the warmest room in the house and the temperature doesn't fluctuate, keep him in there until you get a heater.
I know that when my fish get a fish pale in the face, they're usually stressed. This usually happens to them when I do a water change. He's probably still adjusting to his new home though. But from your photos, he doesn't seem too bad. It'll probably clear up when you get that new tank and heater and when he settles in.
And his tail will grow back. It's still a bit hard to tell in the pics how much of his tail he's missing, but if you keep up water changes and treatment, I'd said two weeks tops based on my own experience. And bettafix is more of a antiseptic, not a antibiotic. It works more for keeping the fins from rotting more. But regrowth of the fins happens naturally, so the bettafix helps keep the tearing from getting infected. Same with salt baths....they work like magic for mild cases like yours.