I noticed what looks like two sets of notches on the cylinder. What's the deal there? Something to do with the DA operation?
Very good observation. Yes, there are two sets of cylinder notches. The rear set engages the bolt, the front set engages a projection from the trigger. When the hammer is down, the trigger is forward and the projection on the trigger is up, engaging one of the front notches. When the trigger is pulled or the hammer is cocked single action, the trigger rotates backwards just like on most other DA revolvers. This lowers the trigger projection and unlocks the cylinder allowing it to turn. As the trigger or hammer continue back, the bolt rises and engages the rear notch. The long vertical cutouts on the cylinder are clearance cuts for the trigger projection, since it is not spring loaded, but is a part of the trigger. It's a really funky system to watch in action. Many of the early S&W 32 and 38 double action topbreaks employed the same system and have the same extra notches on their cylinders. Later models of those guns employed modifications to that system and did away with the extra set of notches. But there were not a whole lot of these DA Number Threes made and there was no follow up model with an improved action.
That's how you can tell the early models of a lot of the S&W 32 and 38 double action topbreaks, if they have the extra cylinder notches. Here is a photo of an early S&W 38 Double Action top break with the same type of action. The 32 and 38 pocket revolvers were very popular and S&W made several model changes as time went on, modifying the action and getting rid of the extra notches. But since only about 53,590 of the DA 44s were made between 1881 and 1913, S&W never modified the action.
Here is a really, really cool 44 DA that I saw at an auction house once. This one was shipped to Teddy Roosevelt just before he went to Cuba at the beginning of the Spanish American War. I believe he actually took delivery of it in Cuba. This one went for big, big bucks, because of the provenance.