1. It was in too dark environment for a while, that's why drooping branches and sparse leaves. But it must have had good light at some time before, since the trunk is straight or someone took time and kept turning it now and then so each side got some light exposure. Drooping branches will not be straight, but any new growth on them will be upright if there is good light.
2. 'Fat' leaves are not a problem, rather that they are spaced far apart (low light is the culprit) and slightly curling under. Otherwise they look quite healthy.
3. Aerial roots are quite common on jades (and some other succulents), so I wouldn't about them.
4. To me, pot is too deep (while quite nice) and maybe little large, but I am sure others may like it and I think it is up to you - do you like it? It also provides good stability to existing plant since it is most likely top-heavy.
Common problem with oversized containers is that they will hold lots of soil. If that soil has poor drainage, overwatering happens easily, and possibly will cause some rotting before it may be detected.
Did you see any roots when plant tipped and lost some soil? There is good chance that they really didn't get damaged.
The plant appears to be healthy otherwise. It is good time of year to do some 'work' on it if you are up to it. It could be pruned quite drastically or just a little - all depends what you, eventually, would like it to look.
You could prune and re-pot at the same time. I would probably just prune it first (since it has some adjusting to do to a new environment) and then repot - maybe within few weeks. It will benefit from being outdoors too - good air circulation, plenty of sunlight. If you prefer to have it indoors, give it best possible light otherwise it will grow similarly.

I would cut every branch about 2-3 inches from the trunk, and let it re-grow all of its leaves. At the very least, get rid of any growth that is not growing upwards. This time next year it will be gorgeous!

What is that soil like? Hard to tell from the picture, but it looks pretty fine/dense. It has a very nice trunk, but the branching is scattered, lopsided. I would agree with the hard pruning in hopes of getting more branching around and up and down that trunk to show it off proper.

I can totally understand the stress you expressed, and if you just took care of it for a year and got to know it, you could make all these decisions then. It doesn't look like it's on its last leg.

'It doesn't look like it's on its last leg' was exactly my thought, too. A few changes in environment (soil and location) and it ought to respond favourably. Just don't expect that to happen suddenly over the next few months, as the plant's partially/largely dormant when it's summer in the Northern Hemisphere.

Man, what I wouldn't do to get my hands on a jade like that. The trunk is perfect, and as noted, the side branches just need to be cut back to remove the saggy stuff. If grown in good light, all the new growth will be strong and upright. I'd also get it in a better potting mix; something more porous and well draining.

Cactusmcharris, can you elaborate on the factoid about jades being dormant in summer? I am in the northern hemisphere, WI to be exact, and mine grow like gangbusters in summer. Mine tend to do nothing in spring and fall, but put out growth in winter, and ever more so in summer.

Joe

Thank you. This is all helpful and what I'm getting is, I don't have to worry too much. The pot is going to go and with that, I'll get new potting mix.

I'm in Wi too. My jades go crazy in the summer as well. Thanks for all this info. I'll post back in a year.

Joe, I should have specified that that advice is true for Mediterranean climates - as many other succs do, C. o. shuts down when it's hot and dry in the summer, as it follows its growth pattern of not growing when it's too hot or too chill. In the northern part of the northern hemisphere this isn't true as the hot and dry are generally not there together.