Yes, there are several ways to frame your wall and it starts with your roof and goes down to the footing. We design from the top down and build from the bottom up.
The building inspector is there to help insure you house does not collapse killing family, guests or visitors.
Framing your wall is the process of transferring all the loads through the wall and ...
You have a chance of finding the studs if you borrow a thermal camera, and it's a hot day with enough temperature difference.
But quite frankly can you find a way into that structure from above? Seeing it for real is the ground truth.
In this case, you don't need to screw the bracket all the way in to the studs. Screwing the bracket in to the wood, which is screwed in to the studs with sufficiently long screws, is more than strong enough to hold a microwave.