Three weeks is very, very young to be away from mom. I'm worried about her not nursing. That is very young. You may want to go to a vet and just see what they suggest.
You really have to stop potty problems RIGHT AWAY. A hard tap on the nose is what I suggest. Both my cats, Motley and Kumar had problems with this. Motley was terrible. I finally had to rub her face in her pee so she would understand. I almost had to get rid of her I couldn't take it anymore. The problem was solved and she is the best cat in the world. The little one was using my speakers as a scratching post. I put an end to that and she is the best and never scratched them again since she was a kitten 11 years ago.
A bit of soft punishment doesn't hurt. It's hard to do, I know, but they will learn from it. They want attention and don't know any better. It's up to you set the rules, teach them and make them understand what is acceptable behavior.
Also, don't over feed your cat. I have friends who have really fat cats. Like chubbers and it's sad because they can't move and play because they are so chunky.
This kitten is still young enough where I believe you can still "train" it.
Buy good cat food. I buy Iams. It's expensive, but it's for the health of my cats. DO NOT BUY FRISKIES!(there is ash in it).
Please don't give up on this little kitten, a little time and patience will help. And I'm sorry to hear of the loss of your cat that passed away.
I can't even think of my life without my two cats. They really mean the world to me.
I really wish you the best and I hope it all works out. Please keep me up-to-date and post a picture of the kitten if you can. By the way, what is his name?
We're reasonably sure it's a "she".
We're calling her "Mitzi" ...
...as in "Itsy-bitsy-Mitzi".