Doors – How should I reattach this garage door bracket

I would expect that behind the drywall at the bottom is a 2x4 plate running horizontally that the vertical studs rest on. The bracket should be low enough that you can use a 2 inch screw and anchor to that.

You may want to repair the drywall before you put the bracket back. Its hard to tell from the photos if the damage is light enough that you can fill it with joint compound or if you will need to patch the drywall. See this question for drywall repair tips.

My guess is that the removal of the set screw (which you called the grub screw) will allow the post piece of the door knob to be removed from the door. A good pull may be required to remove it.

You should look at the door handle on the opposite side to see if it also has a similar set screw. If it does not have any visible means of removal then the knob from the opposite side will pull from the door extracting a central shaft which is extending into the post that you see on this side. The set screw in this case is holding the handles in place from both sides.

Once you get the post off the near side reattaching your knob lever should be as easy as accessing the head of the central screw from the inside of the post piece. The central screw head may use a normal screw driver or may be a six sided hex similar to the set screw head.