What was the mission?

The movie does make a big deal about Space Ranger and finishing their mission. But what exactly was that mission? Why did they go to the planet in the first place? Are they some StarTrek like exploration ship? TitanAE-like seed ship? Are they the only long range space ship around? I didn't quite understand why they couldn't just call for backup or what the hell they were doing out there in space. There was basically no world building as far as the rest of the universe is concerned.

Other questions:

Why didn't the scientists make progress while Buzz was away? They had literally decades to work on the hyperspeed crystals, but only Buzz was ever doing any test flights.

Is time dilation a natural part of hyperspeed or only of those failed crystals?

Where did old-Buzz come from? There was never a split-the-timeline event as far as I can recall.

Why did young-Buzz kill old-Buzz? Old-Buzz just wanted to get them all back home, not exactly a crime that deserves death.

What actually happens at the end? Buzz gets his team and goes back into space. Do they have a hyperspeed crystal now? Do they just travel around the local solar system? Weird to go back to space when the previous conclusion was to settle down. Also given that they managed fine without a space force for decades, that seems like a useless thing to have.

Overall a rather weirdly depressing movie with a ton of rough edges and completely unfit as Buzz Lightyear or a kids movie. Even as scifi movie it is a dud. The idea of time dilation/travel was handled far better in DS9: Children of Time or the anime Gunbusters. This movie felt like nothing really happens, they crash on a planet and it takes Buzz a couple of decades/weeks to accept that. How the rest of the crew deals with it, is never really addressed and just glossed over an a quick montage.