1) SwSh do have game cartridges, but unlike previous generations, save data is stored on the Switch itself, not the cartridge (this also means you can have multiple saves by making new users on your Switch).
2) DLC is linked to the Switch, not the SwSh cartridge (additionally, every user on your Switch gets access to the DLC if you buy it once).
3) DLC does not have a cartridge, and must be downloaded online.
4) The DLC, thus far, is about 1.1 GB (source).
5) You do not need to buy the DLC to have the new Pokémon added to your game; you can still transfer them from Home, trade for them, or join raids hosted by someone with the DLC.