Yes, they (any *A member) can check luggage through even if on different tickets.

No, they don't have to do it.

Yes, A3 will usually do it (was never, ever refused).

Actually, only once they (CM) were not able to do it after having tried for about 10 minutes and they were really apologetic. I was holding different tickets (SJO-PTY, PTY-CUN) but no harm: I had more than enough time to stand on line at immigration and customs only to then go upstairs, check my bag and walk through... immigration. But as there were no queues, I was in the lounge in less than 30 minutes after deplaning.