While not original, I think getting DOOM is good for the Switch. The console definitely needs an online fps, and Doom along with Wolfenstein is a great start!
If you want a new Switch with an included game you’ll have to get The Super Mario Odyssey Nintendo Switch Edition. This bundle comes with the Switch & a digital copy of Super Mario Odyssey. I’ve pasted a list of all the current bundles available now down below.
No the Nintendo Switch can’t play 3DS games. The Switch is only compatible with Nintendo Switch cartridges or digital downloads from the eShop. The 3DS is compatible with 3DS games and original DS games. The Switch is a complete departure from the DS/3DS era for Nintendo. I highly recommend getting a Switch. The ability to swap between a handheld and console is pretty awesome.