Love the game, want to own it on GOG but doesn't it have a hidef version or a button fix or something to enhance it?

To be honest I don't remember anything like that. The controls are quite normal and for the graphics I don't belive there is really need for one. Maybe some fans did it, but...

The game is designed for 4:3 aspect ratio while using black bars on top and bottom to create a cinematic feeling. Displaying the game on modern widescreens stretches the picture which doesn't always look good.

There is a DirectX Injector that offers tweaks for this issue. The files and further info can be found here:
http://community.pcgamingwiki.com/files/file/22-beyond-good-evil-widescreen-fix/

The HD version has never been released for PC, only for Xbox and Playstation, hence GOG's version is the original release.

I was hoping there might be a texture pack available or something. But this is okay, thanks!

It bears repeating that the textures that the last console generation got as "Beyond Good & Evil HD" were the very same the PC version had all along. In that respect, it doesn't get any better than what you're getting here. :)

However, the game's icons were also reworked for the supposed "HD" release, and that's the one thing the PC version never got.

Just play the game.

That will be a good enough experience.

Game dosn't support gamepads so buying xpadder or Pinnacle Game Proffiler would be good idea?

In my experience I've been playing some old games that don't have native controller support with Antimicro. You can adjust the way you like (like making the right stick to work like WASD keys and the left work like the mouse, just to start). It's free and available in Windows and Linux.
https://github.com/AntiMicro/antimicro/releases

The Uplay version support the xbox360 pad, this version was free las year for a month in the Ubi anniversary. Sorry if i was too late.