it's actually pretty complicated! most 360 games were region-free in the first place so it's not a big deal, but i think the region-locked ones may still be locked — sega was the big outlier in my collection. i don't have any compatible OG xbox games to test with.
as for downloading games, you may need to do a bunch of region-switching bank account gymnastics that isn't required for regular XB1 games, but it's doable if you have a card from the region originally associated with your live account. it's probably fine if you're not in my specific situation, ie having moved to another country a decade ago in the 360 era and carried that account over to the XB1.