We had upstairs and downstairs crates while house training. It was just easier. Sometimes I'd just put them in the crate for 5 minutes while I took the trash out or vacuumed.
It's OK to have him on a lap but you have to be good about it. You want to set him up for success which means reducing the chances he will have an accident. Maybe get a sling so you can carry him
Mine are both trained to go outside. It takes a lot of effort at the beginning (and middle... and when you think it will never end...
) but it's definitely worth it! By the way, as soon as they could sleep through the night they got to sleep with us even before they were fully trained during the day.
If you need help let me know