No, they do not have to match. But it is good to have a plan and some strategic overlap so it doesn't look chaotic. Usually the doors throughout a house have the same hardware, for example. And bathrooms and kitchens are entirely separate entities and can have entirely separate finishes. But it is best in a kitchen not to have four separate metal finishes on lights, hardware, faucet, and door knobs. Some overlap, with maybe only two finishes in play, would look more coherent.

Oil Rubbed Bronze and Brushed Nickel pair fine (both are "warm", "soft") for my eye usually. It would be fine to have door hinges and handles in oil rubbed bronze and the upstairs bathroom fixtures in satin nickel.

Absolutely feel free to switch it up. It's totally separate from the upper floor and an opportunity to make it interesting. Personally, I think it would be a bit on the boring side to have everything match.

I'm painting all of the trim in my house, including around every door, a semi-gloss pure white. The interior side of my front door and closet door were painted a glossy black. In the dining room, I'm painting the interior of that door with the same white paint as the trim. It just depends on what look you are looking for. When I paint my bedroom, I'm also going to paint the bedroom doors black since I will be painting my bedroom rose pink and black (this goes with the quilt I made for my room in 2013). The trim in the bedroom will still be painted semi-gloss pure white.