King Marth 64, Please put spoilers on your questions, particularly #2
For this one, what I understand (and worked for me) is you need to do a lot of things throughout the city including:
Do the double battles in Cafe Nah, the Trio *And* Rotation battles at Cafe Yeah, do each job once at the Hotel. Other things that may need to be done: Listen to each painting description at the museum, talk to the older sister of Gym #1 and find out about the different alleyways in the city, go to various other cafes, stuff like that. When you do enough in the city they'll let you in.