How many wires per connector, why is a black wire connected to a white wire, how to install brace bar without attic access?
Think of it this way.  A pair of 3-way switches, together, acts like a simple switch.

So when one of the 3-ways is on a spur like that, it's pretty simple.  The 2 travelers (brass screws) can be ignored, and treat it as 1 plain switch between the wires that are on the &quot;common&quot; (black) screws.
So that would be the white from the /2 cable, and the black from the /2 cable.
That is a classic switch loop.
I very strongly recommend using colored tape to re-mark wire colors so they match their function. In particular up in the lamp box, the black to the lamp and the black *from the switch are both switched-hot. I would strongly advise marking them with red tape.
As far as the white wire being as a hot in the switch loop, that's not an option, Code requires you to mark that black (well technically any hot color will do). So mark that black at both ends.
This makes it easy to see what is going on in your boxes.  Hopefully it will then be obvious that your fourth cable should attach to white and blacks if you want the cable to be always-hot, or white and red if you want it switched with this light.
As far as braced junction boxes designed to be installed through the final hole into existing drywall, they make &quot;old work&quot; fan-rated boxes.   If you put a lamp in a fan-rated boxes, I promise not to tell :)