How to delete chunks of the entire timeline or chunks of the timeline of an item?

I use command + shift + D (Mac) all the time, or Edit > Split. Does a split to the selected take.

Note for stuff like visemese (mouth shapes) to get a gap you have to right-click it and choose "silence."

You can certainly split takes, trim them, move things around in the timeline. It might not be exactly the same is Prem Pro, but all the functionality is there. I just right click in timeline, and look at menu items like under Edit.

The timeline panel takes point to a subrange of a full take, so if you split a take, it turns it into two sub ranges. You can extend or trim either one. The underlying full take still exists internally (you can extend a trimmed take after you shorten it - no data is lost). So splitting a take is like duplicating and trimming in one step.

So for your case i would normally split a current take, trim them shorter, then slide them around in the timeline to remove stuff. Or split a take at the start and end of the part you want to remove, then delete the middle section. (I cannot remember if a fi-ple delete exists - i just drag it around myself as i might do a blend to overlap the two takes to smooth between the before and after to avoid jumps.