How can I check if a compositiones is openned?

I don't think there is any way to do that unless you have a specific item in the comp to scan for or a specific name of the comp itself with the real point being that AE allows you to close the timeline panel or the comp viewer independently, so there is apparently no underlying data structure other than the internal IDs used under the hood that would trigger warnings and such and potentially could be referenced in a script.

Mylenium

I would use the flow chart to open each nested comp I need to look at. That is much more reliable than drilling down through your Main comp.

I would also make sure that my Project Panel was completely organized with folders appropriately named so nested comps (pre-comps) are all in a separate folder for each main comp.

The only workflow that I can think of that will keep the original timelines open would be to open the Main comp that contains the nested comps you want to open (no such thing as sub comps), then click and drag that timeline above or below the timelines you already have open. You will now have two timeline panels in your UI. When you double-click on any nested comp, it will open in that panel group. When have finished with them all you can just close the main comp panel group and you are done.