Okay the other person is onto the right path but I think it's important to understand the underlying reasons for how "backwards compatibility" works on the Xbox.

The 360 used a PowerPC architecture, which at the time was very cost effective at the time. Pretty much most things now use x86, our standard 32-bit and 64-bit CPUs are this architecture. (ARM is another type that we are seeing now).

Now, you cannot run code from one PC architecture to another, even emulated this is a very costly procedure, every call to the CPU, every call has to be translated. Even with emulation this is difficult. (Note how we're still just now getting 360 emulators).

This is mostly why Microsoft and Sony both said no to backwards compatibility, because there was no simple way to take a disc, pop it in, and play.

So after the massive backlash (which they deserved, but also was understandable their point of view), Microsoft created their backward compatibility program. Essentially what they (or developers, not sure who did it) did, was to literally re-compile each entire game for x86, instead of PowerPC. They would then upload the bits to Microsoft, and that is what you download when playing. The disc you insert is purely for checking that you own it, after that you ignore everything else and download the x86 version which is runnable on your console.

So, it stands that backward compatibility wasn't feasible, it still isn't "backward compatible". They rebuilt everything from the source code to run. A pretty massive effort on Microsoft's part and the developers just so we could play old games. Hopefully you see too why I don't blame Sony for not going through all of that, it's a lot of work.

So to answer your questions:

1. Digitally downloaded in x86 with the disc as key
2. No, it is a simple recompile, from the original source code
3. No, since there is only the PowerPC bits on the disc, there is no way to play the game on a newer x86 cpu without downloading the x86 bits
4. They work the same, once compiled for x86 it works for all x86 processors. (Caveats in software engineering of course, but in this case you can assume they are the same)
5. Microsoft went through a ton of effort to get this working, and developers each had to pull up old projects, figure out how to build them again, and did all of this for free. I'm all for hating on Microsoft for a myriad of reasons, but this was a project that had very little profit for them but did the community a huge service. They have my thanks, and Sony has my understanding for why they didn't do it at the time.

AI is not a great source to ask for any sort of factual information, just a heads up. Its good for creating new stuff, like improvising story narratives or something like being a TTRPG dungeon master, though. I just wouldn't ever rely on anything it says to ever be factually true.

Im not an Xbox/Microsoft employee, but I will answer to the best of my knowledge. Some or all may be wrong though.

1. Both.
2. It does not transform the game at all as long as there are no bugs. Usually the only benefit is increased internal resolution, and sometimes increased framerate Everything else is identical to the original release.
3. AFAIK, no. When you install the game it doesn't actually copy the files, it downloads them because some of the files are patched. No internet means you cant download. I think it also needs occasional connection to check. I do think you can play offline but not for 20 years straight.
4. To my knowledge, no, there is no significant difference. I think it probably is exactly identical, actually. It is however, different from how backwards compatibility worked on the Xbox 360. It is significantly improved.
5. Cross-generational System Link. Its cool being able to have one player on Original Xbox and one on Series X in the same game.

The Xbox 360 uses a processor with a different instruction set which means the executable on the disc simply isn't compatible with the Xbox One and Series X|S. For those, Microsoft have worked with game vendors to convert the executable (from binary, not source) to the other platform: "static recompilation". The new executable must be downloaded to play the game.

For the Xbox One, the architecture is very similar to the Series X|S so it can mostly just run it like you can run Windows 7 games on Windows 11 with compatibility tweaks applied by the operating system.

Addendum for anyone interested in learning more: Xbox One should upscale games to around 1080p, while Xbox One X, Xbox One Series X/S and PS5 can upscale to 4K. If you have any further technical additions on upscale differences they are appreciated.