[Galaxy S III Q&A, Help & Troubleshooting] eksasol:
I will try to answer some of your questions:

1 - Is there a cydia app like for android? Or all of the tweaks and stuff are only on the web and on Market (Google Play)?

I don't really use Cydia or jailbroken iPhones, but I used IOS. Android have lots of customizations that doesn't required root or jailbreaking, you just have to know the right apps. "stuff are only on the web" I'm not sure where else you're going to get digital stuff except from the web. You can check the development section. Besides Google Market, there is the Amazon and and SlideMe market and others. There are many different websites and individuals who create apps and mods for Android including XDA, the best way to find what you're looking for is by searching. You can create your own rom, mods and customization. Android is 'mostly' a free operating system.

2 - Is there a way to show my company carrier logo on the statusbar?

The network name is shown in the lockscreen and the notification bar when you pull it down. It's not shown permanent on the statusbar like iPhone is, I'm not aware of a way to enable it, us Android users don't use expensive contract plans as status symbols.

3 - Can I, through any method, change the display app name, in appgrid? Not in homescreen.

Not that I know off, though I don't know why one would want to do that. There is possibly a 'launcher' with that feature, but I don't know.

5 - Any way to have a better adjustement of the volume without depending on apps like Fime Volume Control?

Depends on the phone and what the manufacturer gives. With custom roms like CM9 and AOKP they come with DSP Manager and MusicFX. With the SIII, running a Voodoo compatible kernel you can use "Voodoo Sound" app to improve and tweak the sound.

6 - Any way to make the brightness minimum lower without depening on apps like ScreenDim?

I'm not aware of brightness level per apps. I think its easier with apps like QuickSettings or Widgetsoid and custom roms like AOKP where you can change brightness on the fly by pulling down the notification bar.

7 - Any way to put a youtube video in video popup mode?

Not that I know off, the pop up mode is a new feature only have SIII have. I know there is an app that does it also, but probably not for youtube videos.

8 - Is there a tweak to put internal storage in mass storage mode? I know the one for external.

Not that I know off. I think an alternative is to buy a 64GB version or 64GB SD card.

9 - How can I have a total control of the background app? I have tried advanced task manager, but as soon as I kill one process, for example Brower, it comes back itself within seconds. How can I control autostars of apps?

It is better to leave task managing to the Android OS alone, it preload apps for when its ready to use, the concept is ram unused are wasted ram. I know the MIUI rom have a feature to prevent apps from autostart.

10 - Can I remap the buttons? For example, for a customize action when I long press the home button.

Different launchers have this features, try Apex Launcher for example. But devices with capacitive buttons limits the ability. The Galaxy Nexus have on screen buttons called NavBar that is full customizable. You can enable NavBar on the SIII also with custom roms like CM9 and AOKP.

11 - How do the themes and mods work? On the iPhone the themes were images in folders, and the system images were in certains paths, so if I wanted to change something I just needed the correct path, even to make my custom images. How does all that work on android?

There are many ways to theme, check the Theme section or try searching. Different launchers have different looks. Go Launcher have lots of themes. There are themes patches that make changes to the actual apps that you can apply through CWM. CM9 and AOKP roms have their own theme engine and you can download its themes from the Market. MIUI roms also have their own theme engine also.

12 - How do I customize keyboards? Just installing new ones? Is that the only way?

Install your favorite keyboard, then go into Settings -> Language and Input. The level of customization is depends on the product you downloaded.

Sorry I couldn't help you more, I like my Android to be as stock and slim as possible.