Potty Training Question

The notification will come with time as he learns that to go potty there is a special door he can go to tell you. That will come if your consistent and you have a good schedule with him - the more consistent, the more quickly he is likely to learn how to tell you he needs to go out as he gains control over his bladder/bowels. As a puppy it is not his job to actually notify you because until about 6 months old he can't control his bladder/bowel movements. It's YOUR job to set him on a schedule and to watch him. There are a couple signs you can look for that include circling and frantic movements that you can associate with him about ready to go potty.

The important times that he will absolutely have to go tithe bathroom will be: right when he wakes up from a nap, after running or playing and after any eating or drinking.

Depending on how old he is, a VERY general rule of thumb is that a puppy can "hold it" for an hour of every month they are old (every dog is different and so it is very general). If you have a 2-3 month old puppy, you should be taking him out at least every hour maybe an hour an a half to be on the safe side and again after everytime he wakes up from a nap, after running/playing and after eating/drinking.

If you're letting him run ahead of you when going into different rooms an he's peeing you probably aren't taking him out enough. You also might want to tether him to you do he can't get that far ahead and if he starts you can pick him, carry him outside to let him finish and praise him.

Dont ever punish him for going in the house. He won't associate the punishment with what he did or he'll associate going to the bathroom as a bad thing and he won't go at all or will hold until he can't and you'll end up with more messes.

I agree. The thing about upstairs: puppies (and dogs in general) don't "generalize" well. So, if he's starting to understand that going to the door is the way he lets you know, that works for downstairs. BUT, there isn't that one special door upstairs. So, he doesn't have an option when he's upstairs.

The two keys, IMO, are PREVENTION and SUPERVISION. Prevent accidents by having your eyes on him 100% of the time. If you can't keep your eyes on him because you have to leave the house, or you have chores to take care of, contain him in a crate or pen. Some people find it helpful to tether the puppy to them with a leash so they are never out of their sight.

I agree with Niraya, don't let him run ahead of you to the extent that he has enough time to pee somewhere. Imagine he's a toddler that's just learning to crawl. Parents always just kind of follow the crawling toddler around to make sure that they are ok. Do the same with the puppy. If he moves, move with him.
It's not forever. Hopefully!