History and Workings
(This time the long read first :)
As with his lamp, the phonograph or telephon, Edison didn't invent the printing ticker but improved usability to a point where it was good enough for most uses.
The ticker as used by brokers is based on Edward Calahan's design of 1867.
In this case Edisons improvement was the way of synchronisation(*1). When working synchronized they were celled 'unison' - with one tone. But ticker before Edisons 1869 model had an issue if going out of sync after some time due to mechanical tolerance and electric parameters - called out of unison. So these devices had to be reset after some time. But resets had to be synchronized across all stations, otherwise it wouldn't work. This was done like on full hours or every 15 minute - all with the risk that a station was sending during reset, or missed the reset.
To understand this better a look at the basic ticker is helpful. Imagine a bolt held horizontal by some bearings with a single deep spiral groove and the type wheel (*2) on one side of the bolt - lets say the left. Inside the groove a pin moved left or right, driven by received pulses, turning the bolt and thus the wheel to the next print position. Direction was defined by polarity of the signal. When it reached the desired position, a pulse on the second line made the paper hit the wheel. A third wire was used as common ground for both signals.
Edison addition is an 'open end' to the threaded bolt.
It is essentially the same idea as some floppies used to position toward track 0 - by simply issuing enough steps to reach track 0 from the outermost tracks, with some 'jumping' if already at track 0. Like with a grooved disk and a stop in case of Apples Disk II mechanic.
The new part called 'Unison Stop' was this 'open end', at one side of the groove. For this imagine it at the left side, right next to the type wheel. Sending a left symbols long enough to move the pin all the way across the whole bolt to the left, letting the pin end up in that 'open end' section. No matter if it started out one character or 20 characters right of that position. Given all attached tickers time to perform this will result in making them unison again.
Simple, but extreme useful, as now a sender could sysnchronize all attached tickers before a message got send, making sure that it will received by all the same way. Well, and the same time, most important for brokers, as so noone had an advantage of time or disadvantage of a screwed message.
The Question(s) Asked
What was the communications format of the Edison stock ticker?
Characters were simply pulses moving the pin left or right, depending on polarity (*3). Number of pulses depended on actual position and target position, so if the last letter was a B and the next was a A then one step left was puled. if it was an O instead, then 13 right pulses would be sent.
What about the pulses encoded a particular letter?
Symbols weren't encoded as an independent code set, but relative to the last letter. Kind of a distance encoding.
Electric (current) signalling was by polarity.
Were there (perhaps) different pulses for rotating the letter wheel versus stamping the letter?
Printing is initiated by a pulse over a second line.
Calahan's ticker is based on three wires. One Ground, one for stepping (data) and one for hiting the type wheel (character termination) (*4).
Was there any synchronization between the sending and receiving machines?
Snchronisation is exactly Edisons improvement. Instead of doing this by hand (literally with boys running around the city), the sender could synchronize all attached tickers by sending more 'left' symbols than there were characters.
*1 - Over the years Edison added several other improvements. Nothing as basic as the unison stop, but quite useful things like reduced power need due less friction and alike.
*2 - Later improved models used two wheels, one for letters, one for symbols (*5). They were both mounted on the same bolt. Now the 'print' line was as well polarized. A positive current would make one wheel print as negative the other. Where a single wheel to cope with all characters, numbers and some symbols would need 50-60 positions, making the worst encoding needing 60 pulses, using two wheels reduced it to 30, dramatically improving transmission speed.
Of course using two wheels without shifting them means that all letters are print in one line, while all symbols come in second. So if you ever wondered why traditional displays of ticker data puts stock names in one line and their notation in another, now you know :)) As a favourable side effect it also eliminated the need for spacing inbetween, further speeding up transmission.
And last but not least, it defines why stock names have to be letters only - otherwise they would show up on the wrong line :))
*3 - In some way it's the grand grand grand father of diferential signalling used by USB :)
*4 - Edison did develop and patent a ticker that would work with just two wires first, while still in Boston, but it didn't catch on.
*5 - And yes this sounds much like Baudot's teletype code of 1870 with 'letters' and Figures' and switching inbetween, doesn't it? safe to assume he had the same optimisation in mind.