I'm from Michigan and "no" - a cap and ball revolver/single shot pistol does not require registration. It is considered an "antique".
As far as shipping, no FFL should be required.
However, last year, I was interested in purchasing a percussion rifle (new) from a dealer out of state (not in Michigan). They refused to ship as they claimed they did not know whether Michigan had any requirements regarding that. There are no restrictions unless it be a "local" one and I am not aware of any municipality in Michigan that has restrictions. I pointed this out to them but they were waffling all over the place on it - so I told them to take a hike and I'd buy it from another dealer - which I did and they had no problem shipping it. The only thing that I know of that might stop shipment of a C & B to you (in MI) is if you run in to a supplier as I did on the rifle I wanted to purchase.
A year ago last December (I believe) Michigan did away with the requirement where you had to go to your local LE to get a "purchase permit" prior to going to an FFL to purchase a handgun. Now, that form is handled by the dealer, a background check is called in and verified (unless you have a CPL - concealed pistol license). After the purchase is completed and the fFL paperwork done, the buyer is given copies of a form and you keep one for your records and the other copies are mailed in to the local LE who used to issue the purchase pemits. There is a time period in which this has to be completed and the forms eventually end up at the MSP.
I don't know how it is now that the purchase permit process was changed, but, even though you were not required to register a C & R in Michigan, you COULD at one time if you wanted to. It was an easy thing to do - especially if it had a serial number and then it would be on file if it was ever stolen - they could check the registration and hopefully they could get it back to you.
I had a number of C & R revolvers (saturday night specials) at one time that were of the age they didn't need to be registered (and they never had been). I was going to sell them and I took them in to the Sheriff's Dept. and registered them anyway - then consigned them to a FFL dealer to sell on GB. They were signed over to him as a normal modern handgun would be so that I had a paper trail to him. From there on, I believe he then sold them as C & R or however he handled it.
As an added note - I am in AZ during the winter. If I purchase a handgun here (modern) all I have to do here is the normal FFL paperwork. (A person, if not a resident in AZ, must have a state ID. I have both a state ID and CCW License for AZ). I can legally transport it back to MI. In MI, I can then take it in (or the information on it - make, model, serial #, barrel length, # of shots, etc.) and I was told it would be registered in MI as a "transfer from another state". In the past, I also had several handguns that were never registered that were from estates. Those were taken in and registered as "estate handguns". If a person were to NOT register those guns and they were stopped by LE, they could be charged with "possession of an unregistered handgun".
There has always been a lot of "conflab" on Michigan's registration requirement but it never has really bothered me. If the gun was stolen (and it has to be reported as stolen by state law), if ever recovered, it is a way to get it back to the rightful owner. I'm not saying that it would always happen but I know of some that were recovered and returned - others were probably out of state and in a flea market in the south within 24 hours.