Just to add to the comments already posted:
The local pizza joint is a participant. Does every time I buy a slice of pizza count as a visit?
Probably - but check the earning schedule for the restaurant. Some restaurants only give points on certain days of the week.
If I do that 11 times, do I become a premium member (or whatever they call it) and get 5x points? How long does this premium status last?
it lasts through the rest of the
current year (the year you hit 11 dines) plus all of the
next calendar year.