Sim,1. We intend to stay on a Saturday, will we be able to find a room?
Probably, although many will be full at weekends and especially late at night.
If you can choose a weekday instead, it might save you having to traipse from hotel to hotel late at night with suitcases in search of vacancies.
2. As we are carrying luggages will we be disallowed entry in most?
I don't think that will be a problem.
3. Understand rooms for stay only appear after 9-10pm. How do you book if we decide to rest (7-10) and then stay (till next day?)
I think some places let you pay the short-stay rate until the overnight stay period starts, but it varies from place to place, and you might have to expain that at the check-in counter (if there is one) or on the phone when they call to ask why you aren't checking out after your 3 hours has elapsed.
4. Are foreigners in general not welcome?
No, I don't think that is true at all, but some may not be so welcoming to people who do not understand Japanese, and I have seen several that state this explicitly in their information packs.