I'm at work right now so I don't have access to my library but as a general thing, by the 1870s & 80s, the majority of horseshoes were manufactured (stamped out) in a variety of sizes and types and shipped in kegs (hence the name keg shoes). The farrier would then select a suitable size and either hammer it to fit it to the horse's hoof (cold hammering- a good farrier will shape to fit the hoof, a bad farrier will trim the hoof to fit the shoe) or heat it up and then shape it while its hot (hot shoeing). Hot shoing makes for a better fit but unless you have a field forge, not possible to do in the field.
While a competent farrier could make a new shoe completely from bar stock, it was time consuming and only used when corrective shoing was needed and nothing else would work. With a cavalry company easily having 70 to 100 horses, a farrier would be hard pressed to keep up a shoing schedule even using keg shoes.
Troopers were often issued two spare shoes, one for the front and one for the rear and the farrier will have already hammered them to fit the particular horse in question. If a shoe worked itself loose or was lost, a soldier could replace it but it can be a real trial (and downright harmful to the horse) if they don't know what they're doing. As far as I know, hammers were not individually issued to the soldier and I suspect that they would use another horseshoe as a hammer- both later cavalry manuals and even WWII training films illustrate this method. Also, in later times (and possibly during the 1870s & 80s), either the sergeant or the corporal would have a pair of pliers to pull and cinch nails with.
Owning my own horse, I've had to deal with loose and thrown horseshoes before I could get the farrier out and it was a real pain, especially if the horse has a stone bruise and he doesn't want to stay still for you and is in a cranky mood.
These are just some general comments- I may have missed something or gotten it wrong so feel free to jump in...