I've been using Lightning Wallet for a while, can their Submarine Swaps function increase bitcoin privacy as well ?
I do not know much about submarine swap, but lightning network will help in having privacy as you open and close a channel. I do not know about Muun wallet, but you can use Electrum to open and close channel.
When I send BTC from exchange to Muun wallet, Muun wallet will open a channel. But then does this channel close itself automatedly after the transaction is done ?
You have to open and close the channel manually. Likely Muun wallet do not have a decentralized lightning payment. Use Electrum for lightning network.
Again, use Electrum.