I've been able to reproduce the reported problems. I've fixed the replay going out of sync issue. However the MPR glitchness is a tough one. The thing is that while recording the replay data, I do a fast-seek inside a replay. But LFS doesn't do any (precise) physics during a replay seek, so LFS doesn't calculate the exact positions of every car, but only reports a rough estimation. This is why you see the glitching.
I have now made it so that MPR replay data is recorded real-time instead. This works well albeit slow and provides accurate MCI data, but only for the car that is in focus inside LFS (and cars around it, nearby). Cars further away from the car in focus still exhibit the same glitching as before. This is a multiplayer optimisation and not a bug.
I'm discussing this with Scawen to see if he can help, but no definitive solution has been found yet, although there is an idea.
So what I can do is release a new patch with the real-time MPR recording, allowing you and others to at least record some cars properly. Or I can wait with a patch until we've come up with a better solution, if that appears possible.
If you would like to continue your work / experimentation asap, just let me know and I'll provide that patch. Otherwise I'll wait a bit longer, but I'm not sure how long that'll be until the next LFS Record patch.