If you can't do that, then you need to be prepared for a longer than usual time with crate training and house training. He won't be physically able to "hold it" long enough to get outside or on a potty pad if he's not right next to it for about a month yet -- and even then, it'll take several months of consistent scheduling to help him be reliable in the house.
Crate training at this age WILL involve a lot of crying and screaming. He doesn't have the social skills as a prerequisite to being crated and alone yet. If you're going to crate train him (and you should, so good for you for being prepared for that!), keep working at it. The trick to teaching any dog that it's okay to be in a crate alone is to keep the crate sessions short, positive (treats!) and let him out before he has a meltdown. I can tell you that the result of having a clingy puppy now is likely to be an adult with intense separation anxiety (also very much linked to leaving the dam too early). You will need to help him be more confident...and sleeping on your lap is sweet, but not the best way to do that. If he wigs out at the crate, use a pen to contain him and make him sleep in there for now and then work on crate training during the day as an activity. Confining him in a pen is not torture.
As for the bathroom, probably the best thing to do right now is not allow him free run of the house. Again, he doesn't have the social skills yet to approach new stuff on his own confidently. A gradual introduction to the rest of your house would be easier and less overwhelming for him.
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