I don't know the details and haven't soldered jewelry myself, but as a general principle it might be worth looking at resistance soldering rather than traditional heat soldering. In resistance soldering current is passed through the two pieces to be joined and only the micro gap between them heats up due to much higher resistance. It can produce much cleaner joints with no real heat impact to the rest of the piece. On the other hand maybe it wouldn't work for your materials.
Also an old trick is to use garlic as a "resist" (or anti-flux) to keep solder from flowing where you don't want it.
Slice a clove and rub the juice on before soldering. "White-Out" error correcting fluid also works, as does a thin paste of iron oxide & water.
The resistance soldering mentioned above requires a relatively expensive tool. In my limited experience it works best for small parts like earring posts, and for temporary tack welds to hold items like crowns in alignment when soldering.
In re-reading your post I see noone has answered your question.
Yes you can use easy solder. I use a boric acid and alcohol flux for gold.
It creates a very thin film which is easy to remove with water.
As for minimizing clean-up, That will depend on the setting you use.
If prongs you can use a "peg head" which you insert into a hole and solder from the inside of the ring,
which mostly keeps the top of the ring clean of solder over-flow.
If a bezel you can place your solder chips inside the bezel
so when they melt any mess is inside the bezel, which will be hidden by the stone once it is set.