I'd attach some of those keyhole hangars to the back of the shelf, they're fairly inexpensive. After that measure the distance between them mark two spots on your wall (with a level and the measured distance) and put two drywall anchors in. Then hang the shelf to that. Note that those hangars and drywall anchors aren't exactly high load but the shelf doesn't look to be either.

You're right but to me the particles don't look fine enough. OP should definitely make sure though.

From the texture it looks more like you have some kind of particle board instead of drywall.

Personally I'd unscrew a smaller piece, that'd allow to look for insulation, as well as closer inspection of the "drywall". I also don't think that you'll get a very nice finish on it if this is your first time, at least not for any reasonable effort. Filling the gaps is one thing but getting the whole area flat is a whole other ordeal. Not doing it would leave you with clear filler lines everywhere in contrast to the texture of the wall.