I know we'd all like some scientific actualisation of Star Wars but I mean:
They made noise in space 'cause that's fun.
There was always gravity on pretty much any ship.
I don't really recall any spacewalks so we don't see any instance of 'no gravity'
There's hyperspace since lightyears is a bit of a long time.
Stormtroopers seem very scientifically and inefficiently accurate
At this point I think the Star Wars movies (the oldies) pretty much ignored a fair bit of the science.
But if it was a death star literally put there in our universe, I think there would be a bit of structural considerations for gravity, but not huge due to it being quite hollow. Gravity is pretty strong when the sphere is entirely comprised of dense rock and no air. A mostly hollow sphere of air where air is something close to 1/1000 that of rock (yes, used the density of water lol) is not going to get much of a rollicking from gravity.
Edit: an interesting 'expose' on the moon landings claim one thing: why were the photos so relatively boring? Because they were real and that's all they could get for all the limited resources they had at the time.
Besides technical diagrams from supplementary stuff, the Falcon lands in a docking bay that's oriented towards the first option. There could be some kind of transition point to the second option, but we don't see it and it'd be really awkward.
The Millenium Falcon landed in a bay that was oriented with the N/S axis of the station, but was accessed on the equator. So the interior of the station has a gravity well with "up" pointing to one of the polls.
The surface cannons, surface towers, and trench defenses were all radially oriented with "up" pointing out into space, like you'd expect on a moon.
This also suggests the station was littered with gravitational dead spots and areas where you'd have to carefully transition from one gravity well orientation to another. No wonder everyone is wearing a helmet.
Probably A. It would likely have used artificial gravity just like any starship. Star-Trek has it built into the deck plates but I dunno how Star Wars does it. Artificial gravity can then be dialed in to compensate for the natural gravity of the structure. Which is probably less than you'd think. Without normal gravity effects, the internal air and water pressure will be mostly uniform across the ship instead of denser towards the core.
Same with the matter making up the structure. It'll largely be hollow and filled with air, so much lower natural gravity than an actual moon of the same size. According to official sources (referenced here: https://scifi.stackexchange.com/questions/92401/is-the-death-star-s-gravitational-force-strong-enough-to-hold-an-atmosphere) it's between 120 & 160 km radius, for a probable gravity of ~0.04g. That's not quite microgravity, but still far too low to be walking around in. For comparison Lunar gravity is ~0.166g.
B) if their antigrav is like 2001 a space Odyssey. A) if they're using local gravity (eg built in the gravitational orbit of something).
So depends where it was built.
Either way miniaturization is what they really need to focus their efforts on. Hell, they'd save a lot of space if it were an unmanned drone. Which is also true of Elon Musks schemes to get to Mars.
Judging by their ships, they have gravity generators which are small enough and have a small enough ratio of energy consumption to energy generation to be used in something like the Millenium Falcon.
Which would mean that from an Engineering point of view the option on the left would be perfectly feasible.
On the other hand it does make some sense to structure a combat vehicle as an onion with more mission critical sections inside were they are better protected and less important ones on the outside - you easilly have armour in between levels in that setup whilst in the setup on the left you would need to explicitly add rings of armor sectioning your corridors to achieve the same.
That said, in the Star Wars films we can see that the ship hangars with access to space have a "side" open to space and the "floor" side perpendicular to the radius line of the Death Star, which is consistent with the left side option and inconsistent with the right side one (where the opening to space would be on the top).
to answer this question, ordinarily, i would just reach for the nearest sci-fi game and tell you how they implemented it, but the only game that comes to mind is Space Engineers and that game has both of these