Wood – safely remove stringer in garage

I'm not an engineer, this is just a layman review, but I think you should rethink the main supports. I'd rather see you open the walls and support your ledger boards directly on the existing wall studs rather than lagging into them. I don't think the 3 supports (2x4s?) under each ledger board is sufficient to prevent sheer and since you're lagging into a 2x4, that's a very narrow target to hit through the face of a 2x10 and the existing drywall.

I'd also go 10" on center for the joists, not 16. Otherwise I'd be concerned about the span length of the 2x10's over your car.

I have to ask, who gave you permission to drill into the wall? If you say your spouse, then maybe you need to take some home improvement classes at the adult education night school at your local high school! Just joking.....

But seriously, if you have run into the concrete blocks, you need to use masonry bits with a hammer drill to bore the holes, and than use masonry anchors or specialty concrete screws (tapcons) to secure the brackets to the block. The masonry screws are easiest and the package will tell you what size masonry bit to use. You can use a regular drill, but it will require a lot of pressure and letting off every 15 seconds or so to clear the dust from the hole. Don't be surprised if the screw strips on the way in, very common, it may take a couple of trys. Be sure to drill the hole deep enough. you will get the feel of it. It is not at all like drilling into wood.

I have to mention, if you have drywall over block, then there is strapping somewhere ramsetted into the block. If you can find the strapping, you can use the standard wood techniques. Look for dimples or evidence of drywall screws, that will give you a hint where to find wood. Good luck.