Also, can you elaborate/ point me to info on 2?
Background info: A lambda that spins up stays live for a while, and can continue to handle requests. The function handler is called again, so anything scoped to it, will disappear with the function, but anything scoped outside of the function will persist.
Ergo, move your connection creation outside of your handler.
Realistically you may need some abstraction to properly close and reopen the connection in the event of failure (so you can't have a 'bad' lambda, where something has happened to the connection and you can't fix it without deploying code again, to trigger a lambda refresh), but for the most part it just works.
If you absolutely must hit an RDBMS, be quick about it to free up the connection, and definitely consider if the latency matters that much; a few dozen ms more to initialize and close the connection is going to be a LOT cleaner.
But that said, how badly it penalizes you for scaling connections up depends on the DB.