Normally not, sometimes yes (haven't figured whats the trigger).
When they bump you off to a next flight, you will usually earn miles, as they aren't careful to place you again on an award class.
Generally, under airport control, ticket classes are irrelevant (as in when rebooking they're almost never makpid to keep you in the same class, you almost always get a higher class) unless they are doing a standard change that you would anyway be able to do by calling, but once it comes to bump/delay/cancelled etc. you will get class. Also, being that there are so many classes and only 1 (or 2) mileage classes, chances are not to end up with mileage.
Often they won't be able to place your ff# on the booking.
You should always be able to put your FF# in the booking regardless if it's a mileage ticket, there are other benefits besides mileage accrual (priority check-in, security, boarding, luggage etc.)