Electrical – what kind of cable connector do I use on an outdoor junction box

Fastening not needed

For a short run like yours, you do not need to fasten the wires to the boxes -- the NEC folks have worked this out, and for 12AWG wires, you can run 100' straight up in conduit before you need to put any support in. (This number comes from NEC table 300.19, by the way.)

This looks like it's an old school ceiling box. Note how the metal rod appears to attach to the box on top. It looks like these were attached from below (the bar bends and appears to go under the joist). They still sell a similar product today, with a bar that goes between the joists to hold a junction box.

As noted elsewhere, your best bet is to put a box nearby to lower the number of wires, then run only the needed wires into the box. I've done that myself and it helps a great deal when you have a box stuffed with wires.

If you're determined to remove it, I would cut the drywall back and see if you can detach it from below. Because it looks to be flush with the ceiling drywall, it's impractical to cut using anything but some sort of rotary tool (which would be a massive fire risk). Replacing a section of ceiling would be the least intrusive method of doing it.