You just hole saw out a piece of plywood the same size as your diamond hole saw and then clamp it to the granite.

No, but once he had a talking boat that took him to an underwater kingdom where it turned into a person who was also a king and maybe Zelda’s dad?

Are you saying it’s a 1x6 because the drywall is slightly feathered out around the opening? If so then check again as that might just be the corner bead. A 1x6 trim around an opening would typically not be flush to the drywall - it would tend to crack even with mud and tape.

Is the wall 6 1/2” thick from drywall to drywall? If so then that’s at 2x6 framed wall which would only be used in an interior for either plumbing or structural reasons. If it’s structural then the header above the opening will be solid lumber and you’re golden. Just screw the brackets into the header - that could also explain why the stud finder is finding offset studs since it would be picking up the nails not the studs.

We do stuff like this for toilets all the time, but the blue joists would run through, be doubled, and hang off the joists on the sides, then you would hang the trimmed joist off the hanger.

You would also probably want to block off between joists above the wall which will help the joist that continues beyond the wall to stay upright and help take the weight of whatever is above the wall.

Make sure that there is no structure above the piece of the joist you are taking out.

Re-route that electrical wire to the other side of the joist so that it doesn’t get in the way of the hanger and header.

If that sounds beyond your ability the get a professional to do it. And since they will cost as much as a new projector, then just buy a new projector that throws the distance that you need it to.

Before any of that: read your projectors manual in case you’re missing something that will allow you to adjust the projection to the space you have.