It is more difficult than assumed to find easy to set gemstone mountings. Most dealers want to sell finished goods and those who supply good to high grade mountings without stones sell only to established jewelry businesses. Such mountings require a trained jeweler to do the actual stone setting and finishing of the jewelry.
There are some companies that what they call "easy set" mountings. Without using the means a professional jeweler does, meaning lots of training and tools, this is not a bad way to get a feel for the work. You will be able to produce simple, though not elegant, jewelry in silver or gold. Some of the settings are not as easy to "set" as might be implied. Be ready to make a few mistakes and practice first with silver on your least appealing gemstones. When comfortable with the methods, go onto better settings and better stones.
What do I think of these sorts of settings and the "snap-tite" settings? Frankly, since I do work totally from scratch and by using purchased settings of high quality, I am not very impressed with the "easy set" and "snap-tite" settings. I suppose many years ago when I first started I would have seen these as a life saver, not knowing at the time how to set stones the professional ways. One sock truly does not fit all and sometimes fit will be a problem if stones have thick girdles(waists)or are cut with very deep pavilions.
The finish of the final product will not be like I would do with final finishing to a perfect surface but that cannot be and should not be expected in this sort of setting. The settings do look pretty good overall when you take the time to practice first and then do the best you can with better stones. No one is going to look so closely at the details of the finish as I would. Most regular non-jeweler folks who do like jewelry will likely oooh and ahhhh at the pretty jewelry you have done.
Prices are higher to the hobbyist and selling structures are slightly different. In either regard, the products are good overall and the catalogs provide some limited "tips" or clues to doing some kinds of jewelry work. There is not an on-line catalog but you can order catalogs from the website.
Now, working through your question, you asked: "I was thinking the solitaire mounting would be easy to find and should be able to get from the jeweler, but the reproductions will take some hunting. Is it cheaper to supply the mount myself? It there more a risk for damage and a poor setting job if i supply the mount myself?"
"The Emerald Cut Stone". A simple solitaire type would be fine for this stone. The problem often encountered with colored stones is the "cut", because often the stones are cut with a fairly deep pavilion (bottom half of the stone). If you simply purchased a setting for a single emerald cut stone of the right mm size, the setting might not fit due to depth of the gemstone.
In your case the depth of the emerald cut is not out of line and will likely fit a standard setting, but it really should be fitted to the mounting by hand to know for sure if modifications are needed. A "basket style" setting would likely be best. This has a rim of metal around the prongs just below where the girdle (waist) of the stone will fit in the setting.
"The Antique Style Mounting for a Square (cushion) Stone". This setting will likely be difficult to locate. Many mountings are available with the openwork, filigree antique look you desire. However, to find on for the size and shape of stone you have presents a problem.
I seriously doubt you will find such a mounting available by your own searching; then, without being a jeweler and knowing how and what modifications can or should be made to a mounting, you really may not know if the ring will actually work for your stone. This one is also best left to a jeweler. A jeweler will have a much wider range of styles and qualities from several suppliers.
The jeweler if competent should be able to set this square cushion well enough in a proper mounting or purchase an antique style ring base and add a proper setting to the base. I do recommend having a jeweler do the work and supply the rings for the two stones you particularly like. The jeweler will have no reason to do a less than normal quality job on a setting you provide, since the reputation of the jeweler goes out on your hands whether the setting came from the jeweler or not.
But, since the setting for the square cushion may be a hard one to come by, if you find a mounting believing it will work then the jeweler has to do extensive reworking to actually "make it work" the overall work may not be what you expect in final looks, even if technically well done. Also, if a setting is purchased from a jeweler, the sizing to your correct finger size is generally "not" at an extra charge to you, it is part of the sale.
Going up two sizes from an 8 to a 10 will not adversely affect most settings, exceptions being rings with mostly recessed or hollow areas inside the ring which do not like to bend and round out easily and cleanly. Likely the jeweler cannot get the ring ready made in 10 anyway and part of the plan is to size to the correct finger size.
Overall, the sizes of the two stones are nice but not extreme.The sizes should not be a problem in finding mountings. Another reason to have the jeweler supply this setting is "turn around time" on the job. Sales take precedent and during busy seasons that is very true! A job with only labor charges and no other sale involved will sometimes be one of the last to get done during very busy times. This is just a part of business.
"Rhodium". Years ago ladies bought white gold and wore it and had it buffed once in a while and never thought about a rhodium plate to make all whiter. Now, that appears the thing to do. With some white gold with a more yellow look the rhodium is a good idea but often a simple buffing will make the ring look just fine, even if not as white as rhodium. As for replating, the stone does not need to be removed.
Porous stones like turquoise and stones affected by acids such as pearls must be removed or otherwise protected from the solution. Topaz and most faceted colored gems will do just fine as set and no stone removal is needed. The solutions do not affect the gemstones.