I feel your pain X2.
I have a 2520 and mow 20 degree side hills. I have both the 62D and a 72 inch rear finish mower, but primarily use the RFM because I feel it delivers a better cut and it's easier to trim under trees, etc.
My property is so hilly that I have to mow in 4WD all the time. The RFM is heavy and tries to pull the rear of the machine sideways on the hills. I'm not complaining, just giving you some background as to my situation.
Anyway, although I knew I'd need ballast weight, I chose not to load the tires and went with wheel weights (the maximum 3 per side). I also keep the loader on (without the bucket) and feel that the extra weight helps.
The upshot of it is, it works, but just barely. The machine still slides sideways on the hills and will still break traction fairly easily if I'm not real careful. I haven't run my 62D in awhile, but didn't notice a lot of difference, other than less of a tendency to pull sideways.
On the plus side, I can report that the machine has always felt stable in this environment, however, I never let my guard down. On the steepest areas, I take it very slowly and am always ready to take corrective action if it acts like it's going to tip.
I have never tried carrying the weight box when using the belly mower, but it would probably help traction. However, I wouldn't want to carry a lot of weight behind the rear axle and nothing on the front. If/when I try it, I would probably have my loader frame on or suitcase weights on the front rack to keep the front end from getting light.
Generally speaking, I'm kind of leery about using the ballast box as weight for traction on hills because I think a good portion of the weight may wind up above the axle line, which isn't the best for stability.
I can't answer your questions except to say what I would do and there are many people on here who are more knowledgeable than me.
1. I have tried mowing with/without the loader and (at least with the RFM) I prefer to keep the loader on and drop the bucket off (for maneuverability).
2. Loading the tires is probably a good idea, however, I personally chose not to do so because I did not want to deal with the issue of losing the liquid when/if I got tire punctures.
3. As mentioned, I haven't tried the ballast box when mowing, but have given it some thought and would probably try it, but not with full weight. The way I have mine set up, I can vary the weight from 400 to 800 or so lbs and I would probably try running with the minimum amount in the box at first.
I dont' know if any of this helps, but the message is that you may need to experiment a little to see what works for you.
Incidentally, the picture on the left is on the 20 degree side slope, although it doesn't look that steep.
Wishing you the best of luck.