Multiple "ins and out" from the same sequence

In the project panel, right click the sequence > duplicate.

Open the duplicate sequence, make cuts and delete the parts you don't want.

Fill the empty spots with black video if needed.

I like editing that kind of thing.

I would drop the clip into a sequence. Then go to the first spot to 'save' and press <Q>, then go to the end of that clip, and add edit/cut (<Ctrl-K> by default), then just do that again:

Go to the inpoint of the next section to keep  and Press <Q> then the end of where that clip ends and Add Edit. <ctrl-k>.

You cannot. You could either cut out what you don't want, leaving your highlights. Or copy/paste your wanted highlights from one sequence to another. Or make subclips of your highlights and then put those into a new edit (but that's the long way). Or open you sequence in the Source monitor and edit your wanted highlights into a new sequence. 

Multiple ways to achieve the end result.