I have found a way to bundle a minimal version of TeX Live inside the AppImage. The resulting AppImage can output PDF files without any existing LaTeX installation in the system and is 149 MB in size which I think is still reasonable.

If needed we could also have a script check the system for a pre-existing LaTeX installation and if it exists (e.g., latexpdf is already on the $PATH) use that instead.

Here is the recipe, I am using your AppImage as the basis and bundle the minimal version of TeX Live:
https://github.com/probonopd/AppImages/blob/master/recipes/texstudio/Recipe

And here is the AppImage:
https://bintray.com/probono/AppImages/download_file?file_path=TeXstudio-2.10.8.tl20160405.glibc2.4-x86_64.AppImage

There is one remaining issue which I couldn't resolve and would love to get help on:
If you open the AppImage and try to run pdflatex, then the pdflatex binary is not found (although it is on the $PATH, see my custom AppRun script). However if I go to Options -> Configure TeXstudio -> Commands -> PdfLaTex -> Select Program -> Press OK, then it starts working.

What do I need to do in order not to need the manual Options -> Configure TeXstudio -> Commands -> PdfLaTex -> Select Program -> Press OK?

Hello!
I'm using the AppImage of TeXstudio 2.12 and can't change the appearance of the gui
When I run it, the appearance is not the dark one.
So I tried to change that in Options -> Configure TeXstudio -> General -> Appearance -> Style, but all of them look the same with the color scheme "Modern".
Is that a bug?

PS: I don't know if this is the right place to inform that, if it is not then excuse me

The binaries inside the AppImage should be stripped to save space.