... is really mediocre. Is the any other way to surf the web? Does something like a homebrew alternative exist? Or is there any other way to add (browsing) software to the PS Vita's "LifeArea" - Surface?

Unfortunately, at this time, there are no homebrew web browsers and AFAIK no one is working on one yet.
I don't believe anyone has made one yetis there any update for a custom web browser for the PsVita?