This is the rule:
Once approved, the biographies cannot be dropped or switched for a month.
1) So the month starts when your bio (or bio's) gets approved.
2) You don't have to wait a month to make another bio when you fail a test as the bio didn't get approved (because you failed) and the month starts (as said in my first answer) when your bio is approved. However when you fail a test often there is a time limit on it for that specific bio. For example when you fail a Pein test, you can't try to make a Pein bio and take the test again for some months, but this doesn't apply to other bio's.
3) Not really sure about this one. You must really suck and/or must be really hated if your bio got killed 3 times in one week >.> But I would say that killing a bio would mean the same as dropping the bio, so you can't make a replacement for that bio till the month is over, however you can have 2 bio's so you should be able to continue with your 2nd one. In the case you don't have a 2nd, you can just make a new one.
But as I said I'm not sure about the 3rd.