I am new to electronics... Can a logical gate treat negative voltage as a logical 0 and the ground voltage as logical 1 ? Or should the voltages allways be between 0 and +12V ?
1) 0 and 1 are just to make things easy for humans.
You could also call a certain voltage range False and another one True.
You could even call them Bert and Ernie if you like but that's confusing for humans, also 0 and 1 is more compact to write down.
In the same way, if you wanted a circuit where + 5 V is 0 (zero) and 0 V is 1 (One) then be my guest. It's just a naming convention. The transistors don't care either.
2) A circuit with a +12 V and a 0 V (ground) has +12 V just because it was chosen to be like that. You could also connect that + 12 V node to ground instead but now, since it's ground, we don't refer to it as +12 V anymore because that is confusing. Now we call it ground and then the node that was previously ground is now called - 12 V
In this circuit the traditional 1 (one) would be 0 Volt and the traditional 0 (zero) would be -12 V.
In the early days of logic chips some used a common positive so all voltages in the circuit were negative ! A bit confusing now since almost no one does that anymore but it is possible nonetheless. Ground is just a reference the designer chooses.
How about if you negate your already negated output?
If you put in 0V, you get on output of first negator (called N1) logical 1, and if you negate that, you get out logical 0 on N2.
If you put in 12V, you get on output of N1 logical 0 and if you negate that, you get logical 1 on N2.
By that, I don't see really much of an advantage.