Direct debits are moved over and are covered by the switching guarantee, so you should never be out of pocket if there are any problems.
They won’t get switched - only your current account can be switched.
For a current account switch, you start the switch with the bank you are switching to (Monzo) and they handle the whole process.
If you have an overdraft with Barclays and are using it, you may have to pay that off first. Otherwise, switching all happens in the background, once you’ve requested it.
Unfortunately, current account switching won’t help you and Monzo don’t do mortgages anyway. If you want to move your mortgage, that’s a whole different conversation…