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.
I would point out that you really do not "deposit" any coin to a lighting node.
You open a channel with another lightning node, and as long as both of you have the relivant keys, the state of the channel can change provided both of you are in agreement. If the node that you opened a channel with losses access to its keys, it will be unable to ever update the channel state, and you will eventually need to close the channel.
However if
your node losses access to the relivant keys, your channel partner will need to eventually close the channel. If you have also lost the keys to the address the closing transaction will be sent to, you will have lost access to that coin, as would happen in any other case in which you lose access to your keys.