Certainly. This photo from my Elna sewing machine manual shows the elastic thread loaded onto the bobbin, and what the result looks like.
This next photo shows a simpler way of doing it which may not produce such even results, if the elastic slips along the stitching.
Caution: I have used neither, but I know the Elna manual is fairly reliable.
Yes, that stuff looks like what you need for the bobbin version. You could use it or somewhat heavier 1 mm 'hat elastic' for the second version.
However, I think you should also consider using some of this:
or maybe some of this:
in your creation :-)