I don't understand all of your questions, so I will give you my feedback based on my understanding.
What Number would then stand in the Register T3 every tick?
It is the resolution at that in your case it is at 12.8us
How is the period or where is the period applied to?
In what purpose?
The period is the full 16-bits of the register so
where goes the Input frequency into?
This is the clock frequency is the rate in which T3 changes (increment or decrement)
What does the gated Mode or how does it work?
There is a section in the manual that describes this. In T3 Gated Timer Mode, the input clock of T3 is gated by the external input pin T3IN. You can control the level at which it gates the clock (think of it as run enable controlled by a port pin)
What Number would then stand in the Register T3 every tick?
It is the resolution at that in your case it is at 12.8us
How is the period or where is the period applied to?
In what purpose?
The period is the full 16-bits of the register so
where goes the Input frequency into?
This is the clock frequency is the rate in which T3 changes (increment or decrement)
What does the gated Mode or how does it work?
There is a section in the manual that describes this. In T3 Gated Timer Mode, the input clock of T3 is gated by the external input pin T3IN. You can control the level at which it gates the clock (think of it as run enable controlled by a port pin)