Can I safely hang pegboard between studs, rather than across them?
Yes, you can place pegboard between studs, but as you stated, you need some clearance behind the pegboard. You could use 1x1 attached to the side of the studs to screw into. This will give you better holding power than brackets alone.
When I did mine, I used 1x2, but I screwed them in across their width, so I'd have more surface area to screw to. My studs were not exactly spaced and straight so I needed a bit more area to attach.
No glue in my opinion.