Changing the duration of the playback time of whole animation

I'm new to Adobe After Effects and I'm working with Audition, Premiere and After Effects. In all three application the control seems to be quite different in order to manage to set the time scale of the part of the animation I'm acutally working with.

In the screenshot below you can see the After effects project I am actually working in:

 I am now working in the middle of the project and like to know how to instruct after effects that 

- the whole animation is played

- that a specified part ist played, having controll about the loop lenght of this playback part

I really searched google and youtube before after I tried to manage that about nearly an hour now.

I would expect the controll of this feature somewhere here, or on right mouse click.

It's frustrating, due to the fact that in Adobe After effects 

It's all about the controll of that red line intervall, which seems to be at the moment limited to about 2 seconds, than it loops to the beginning again:

I also checked => composition => composition settings => duration:

This has no effect and seems to be the wrong area for this control...

It would be very kindfull if someone of you could give me and others who might also search this a understandable solution.

Thank you very much!