Hi That back looks to me as though it can be removed. look round the perimiter for an indent for inserting a case knife if none visible you may need a case crab like the one attached as some of these cases are tight. can you post close ups of the case edges it may help.
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