What would happen if the lightning node you chose, lost its multi-sig key after you deposited your coins?
You always have your channel closing transaction after depositing coins. And that means if someone is not cooperating, it does not matter if that node lost the keys or if is simply offline, you can always close the channel and then open another one with someone else.
I obviously know the answer, I just want to ensure that it can't happen in my future deposits.
Every time you deposit something, the closing transaction is constructed and signed upfront, in this way you are always protected when another party stops cooperating.
Because of Segwit, it is possible to create the first transaction, create the second transaction taking inputs from the first, and sign that second transaction without signing the first one. That means transactions are signed in reversed order than they are created, so the final signatures for the first transaction are exchanged where everything else is ready.