Restoring shine to the old paint may or may not be possible, depending on what is wrong with it (eg road film, industrial fallout, chalking, or what have you). Way to know is by buffing a small section, with a good cutting compound (light or med grit). If ya got a machine with a foam or wool pad, run it at 1500 rpm. If not, do it by hand but don't use T-shirt rags, because they are too thin and scratch. Use a rag that's got some body to the fibers. If it comes back find someone whom will buff/cut the rest (not polish). Detail shops won't do this for the price of detail, or really don't want to do it at all, because they can't charge enough to make if worth while.
To cut & buff a side is worth $200. to $300. to do it right. Detail shop would probably use a product to give it a "temporary" nice shine, to get it out the door, then deal with you later when it dulls out again (or if they are reputable, may tell you up front).
Just remember buffing is thankless, nasty hard work, that painters hate, again because no way customers would ever understand paying, what it would take to make it a profitable operation. Detail shops if they say they are going to cut & rub a side for less than the above: I'd go back (unannounced) to see for yourself it's what they are really doing.
Whatever you do, don't try to buff it with an random orbital type machine, because they are not designed to cut, only polish. If all it needs is polish, however you'll know by testing above, then yes fine to use the orbital.
As far as matching paint, there really is no such thing except with a very few colors on a very few cars models. The reason is because after market the paint materials (pigment, binders, extenders, plastisizers, surfactants and so on) are entirely different than what is used in the plant. Same for painting equipment, not the same thus can't atomize the same.
Color matching is the art of creating the illusion of match, via blending techniques. This is not to say one should not adjust the color (tint), just that you can only get it so close (unless a painter makes the time to spend as much time as necessary screwing with the color, and wasting time that could be used on another job. It can take literally days on certain colors & situations, to get a color as close as is possible, and quite typically several hours) Most will dial it in close enough for a "Blendable Match", or if it's not too far off forget any tinting, and rely solely on blending.
There are a great many color formulas that are blendable as mixed, but there are also many that are not acceptable, to blend without first tinting. There are exceptions to everything, but not many in what I describe above.
Credentials: Professional painter for approx 40 yrs. 15 of them, as a tech guy for a major paint manufacturer, training, trouble-shooting, & new product development.