I wonder if part of the reason he bites is because he doesn't know if he's going to get smacked in the face, and he rather bite before it happens. You didn't say how old he is. My puppy is almost 15 weeks and he bites (well tries) when you give him praise and I think its because he just gets so excited. So i started to tell him no bite, lick..and make a big deal out of it when he licks, and he doesn't try to bite that much he will now lick fingers when he gets excited.
When my first puppy would go outside to potty he would do the same thing and it drove me nuts, because he would pee right at the door frame before we went in the house, what helped me was taking him for a long walk, and not letting him play, if he started to do all those things you mentioned then I brought him inside for ten minutes and then took him back out, and kept saying go potty, I also think it helped that I made the leash a little shorter.
He is going potty on the carpet over and over again, probably because he still smells where he went. I always did this when my first puppy had an accident.
Dog Urine: Clean & Remove Dog Urine in Carpet FREE Recipe
Or you can get a product like natures miracle.
As for the whining in the crate have you tried the kong toy and filling it with treats? My first puppy use to love that, not so much with treats because he didn't like treats to much, but he loved frozen water inside, and I froze small pieces of fresh fruit to go in there, or mixed a bit a baby fruit, with shampoo.
Or something to chew on like a bully stick?
I'm not sure about the time out/punishment, but I wouldn't use the crate for that reason because your baby will only see it as a bad thing and wont want to be in there, because there always going to think they are in trouble.
And lastly don't give up, when I had my first puppy I was so stressed wondering why he didn't go to the bathroom outside when I wanted, why he pee'd on the floor etc..and I know now that stressed him out big time and he was feeding off of my emotions.
Now that I have my second puppy I don't stress at all, he came home with me at 9 weeks and he hasent had an accident yet, and if he will and I know its going to happen eventually I wont stress, just clean it up, and correct him if I see it.