I never tried removing the rear wheel without removing the caliper first, so I can't tell you that it absolutely has to be removed. Having said that, based on my experience I highly recommend that you do so. First off, you only have to remove two caliper bolts, and it slides down fairly easily.
Second, removing the caliper gives you more room to work with. If you don't remove the caliper first, although there may be room to remove the wheel, it is still a tight space. As such, you do run the risk of scratching up your rim at the least, and possibly damaging your rotor at the worst (though not likely, it is a possibility). Additionally, with the caliper removed, it makes installation so much easier as you don't have to try aligning the rotor with the opening in the brake pads, while at the same time trying to align the Wheel's center axis with the holes in the elliptical adjusters to insert the axle.
In regards to your other question, that is all that is necessary to remove the rear wheel. You don't have to remove the rear caliper bracket, nor anything else.
Here's the actual steps I typically follow when removing my rear wheel.
- Remove the rear brake caliper, and tie it up out of the way.
- Use a sharpie or alcohol pen to mark the current position of the swing arms index mark. This will facilitate resetting the chain's slack on installation.
- Break the torque of the swing arm pinch bolts and rotate the adjusters to loosen the chain. You only need to rotate one side, as long as the axle nut is still torqued down.
- Tighten the swing arm pinch bolts. They only need to be a good hand tight at this time. This will lock the adjusters in place and maintain wheel alignment when reinstalling the rear wheel.
- Remove the two rear axle snap rings.
- Remove the axle nut and slide the axle out (to the left side of the bike). You should be able to easily push the axle out. If not, it is likely that the weight of the wheel is causing the axle to bind. Simply apply a slight upwards pressure on the rear wheel and try to keep it level. This should relieve the weight from sitting on the axle and allow it to slide free. If it still doesn't move easy, or moves but stops, more than likely the rear axle is still binding due to the wheel being out of alignment with the adjusters. Simply check the right adjusters axle nut hole to see if the wheel is aligned with it and adjust the wheel as necessary.
For reinstalling the rear wheel, simply follow the above steps in reverse order with a few provisoes: First, once you reinstall your axle nut, make sure to torque it to spec before loosening the pinch bolts to tighten the chain. Second, once the axle nut is torqued, go ahead and loosen the pinch bolts and reset the chain slack by by aligning the index mark you made earlier with the index mark on the swing arm. Snug the pinch bolts back down but again only hand tight. Now reinstall the rear caliper, and torque it to spec. Put your bike on its side stand and double check the chain slack. Make any necessary adjustments and then torque the pinch bolts.