When I put the piece back in the hole I cut, what do I use to re-attach it so it stays stable? (so it won't fall back or forward out of the hole.)
Glue tabs on to the back side of the hole, then glue the cutout to them once the glue used has had a chance to cure. You can probably just use superglue/CA to attach these tabs as they don't have to take much strain, but it'll be stronger if you use PVA, polyurethane adhesive or epoxy.
Note: you can use more than one glue, with the main glue spread over most of the surface for strength, with dots of superglue just to provide initial hold. But many times a filler panel like this is just held in place with masking tape until the glue dries.
If this can be done, what do I use to fill in the crack of air that will be around this piece that was cut out, after, it is re-attached?
Any kind of filler might work. It doesn't have to be a wood filler necessarily, depending on where this is, how visible the surface is and if visible how important it is for it to look a certain way — if this is in a utility area you might be fine with white filler material. And obviously if you'll be painting the OSB the filler colour doesn't matter.
On the commercial side any "plastic wood" product, Durham's Water Putty, spackle (UK: Polyfilla or similar) which can either be used plain or tinted, or even silicone or acrylic caulk if you wanted.
On the home-made front you can make decent filler material, with any consistency from sloppy to quite a stiff paste, from wood dust1 and glue. Epoxy is a good choice for this because of its strength2 but it works fine using any standard PVA glue, white or yellow, although this DIY filler shrinks a little upon drying and tends to end up a darker colour.
1 Even sawing and router waste with little flakes in it, not just pure dust, would be fine for this fill.
2 Any ol' cheap 5-min epoxy, even stuff from the dollar store/pound shop, is perfectly adequate for this purpose.