Hi The right weight is as explained and drives the time side gear train the left is the strike side and only releases either on the quarters or on the hour depending on type of clock. fit the minute hand and move it round the dial to determine the strike pattern stopping at the quarters and watching the front plate mechanism for the release. This movement will probably have a count wheel with the slots in governed by a set of levers/wires as per the attachment.
If your app works decently I'd say a beat error of 3.1 is excessive. You would have to remove the balance and rotate the collet. I believe I read somewhere that a beat error of about 0.8 ms is acceptable for a watch like this. Watch Repair Channel has at least one video demonstrating how make the adjustment of the collet. Perhaps you'd want to practice on a scrap movement first. As for the rate I see this movement is fitted with a swan neck regulator, so hopefully you'll just have to turn the screw clockwise to slow it down.
The right side weight is for the time and the reason its falling is the verge (the part you said fell off and you have pictured) is not attached. The verge meshes with the escapewheel to cause that tick tock sound and regulate the motion works to keep time
I’ve had this pocket watch movement for years and finally gained the muster to strip and clean it. Turned out relatively well but now to regulate it, I have a cheap phone Timegrapher app that says +76s with a beat error of 3.1. The only lift angle info I could find was 44 but nothing definite. I’d love to regulate it myself so if you guys have some tips please share. I’m not adverse to taking it in but probably won’t be able to until winter