You have too many questions in one, which is generally bad form on SU. But since this question is aging - I will try to start you in the right direction:
How do I autohide the panel in Mate?
Mate is a fork of Gnome2, so the same methods should still work.
- Right click the panel
- Properties
- Autohide
If that fails, you can launch mateconf-editor (terminal or from System Tools, IIRC).
- Navigate to Apps > Panel
- Autohide checkbox
Or how do I use it more efficiently? Maybe there is a way to display "open since: time" or add an "always on top" button beside the "- + x" buttons for minimizing, maximizing and closing windows.
That depends on how you work, so I can't speak much to this outside of using mateconf-editor for your windows to see what can exist in the top bar. Last I checked for them it was Menu, Min, Max, Close.
Or how do I set the menu button and tab bar from Firefox in the panel, like in Windows?
You'll need to be more specific, but assuming you mean something like this:
I'm not sure that is currently supported by Mate - but I am not fully sure there.
Would it be possible to make an extension for the Mate desktop, and where can I get started with it?
It's Linux... Of course you can. However - the question is more how flexible is the Mate API - or will you need to re-write anything of theirs?