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.
Edit:
What happens if the Mk node fails at the time of transmitting this information.
Then, the whole chain is cancelled and another route is tried.
Nodes A, M1, ..., Mk-1 will assume that the transaction is completed. And the nodes Mk+1,...Mn,B will remain in uncertainty.
They don't, because nodes don't blindly trust that LN internal transactions (with HTLC and millisatoshis) are final. They are final only when closing channel transactions are updated for each node.