No, don't let him get away with yipping and yowling in the crate.
One way to teach him to be quiet in the crate is put him in the crate, let him start his yipping and yowling, and as soon as he is quiet for about 10-15 minutes, CASUALLY let him out (don't make a big deal out of it) and praise him gently.
I also find sometimes that covering the front of the crate with a towel helps, too.
If neither of these seem to be working after some time, resort to this: when he starts yipping and yowling and DOES NOT STOP, walk over to the crate and give it a hard smack on the top while using a command like "Hush!" Then, like in the above, if he is quiet for 10-15 minutes, let him out.
Later on, if you feel he is ready to graduate from the crate, you could put a baby gate in front of the bathroom or kitchen instead of closing the door so that he can still see what is going on (and hopefully it will save your door from being torn up again).
Just remember not to let him get away with things because small dogs are the world's worst at turning into snotty brats.