Do not launch the game yet. Immediately install the UPDATE NSP. The installer should recognize it as an "Update" for the existing title ID.
The DLC expansions significantly broaden the game, though they vary in appeal: A Hat in Time + Seal the Deal (Switch) Review A Hat in Time Switch NSP UPDATE DLC
A: Not officially. Through homebrew tools like Checkpoint and manual hex editing, some users have converted PC saves to Switch, but it is not straightforward and can break DLC flags. Do not launch the game yet