About a year ago, I completely rebuilt a violin. I had no luthier experience, just general woodworking and carving. Here's some advice for your project.
If the pegs are predrilled, you may have some problems if the peg box holes have been reamed to large for your new pegs. The peg's string holes might enter to far ( or not far enough) to keep your pegs tight once the string starts to wind on them. It can cause a cross torque that always makes your pegs want to loosen up or slip. If this is the case, fill the original peg string holes with epoxy and "carefully" redrill the holes in the proper place (perpendicular). If the end of the pegs are a lot lighter then the rest of the peg (after you cut them to length), rub them rapidly across pink rubber eraser to burnish the end. It will shine and polish it nicely with little effort, so no need to stain/dye/paint them.
Insure that both peg box holes have the peg sitting securely. If they aren't, additional stress with be placed on the pegbox and your tuning with not go smoothly and your string will ride one way or the other up or down the peg.
It looks like the end nut is worn too. This can cause buzzing and hard to play strings. I found, for me, the end nut was the hardest thing to make. It seems easy enough, but it took multiple attempts to get it right, and still, I suffer from 1 slot not quite square. The angle is extremely important on the back side. Pressure there at the wrong angle can damage your string and cause the strings to feed wrong into the pegbox.
Somewhere here, I had images of the whole project, but I seldom post anymore. I just troll and read and play along in my own secluded little realm I call home. This post really caught my interest. hopefully you have more experience then I did and your project goes smoothly. Ask away if you have other questions. I know for fact that several really talented people tinker on their instruments and could add loads of insight.
"I find your lack of Fiddle, disturbing" - Darth Vader
Learned a couple of things.
I use reversed spring clothes pins for a mute (clipped on bridge,,both sides). I put them at the bottom of the bridge, which took away the worry about the bridge falling. I still have to pay attention, but the worry is gone. I still put a cloth under the tailpiece, just in case.
After putting on the tailpiece, the nylon strap stretched and I was not close to the 54mm for afte rlength on the strings, so I took it off, tightened the nuts some more and reinstalled. Then while playing I heard a pop and realized that the threads on the strap were giving way. It makes me understand that once the nylon has stretched I can not go back and retighten. In other words I get one shot at it. So I took it off, put on another, tightened it so that the wood just barely had enough room to go over the little insert and after stringing I had 52 mm after length,,,close enough.