I personally just tested this out by pairing parts together with someone else, and I experienced no rhythm issues.
When I composed the parts themselves, they weren't cut off either.
I believe you are experiencing a mix of bad MML being copied to your clipboard alongside computer lag.
My computer's currently broken though, so I'm not sure if I can retrieve the original MML. Nonetheless...
..There really shouldn't be an issue. I'll show you footage if you need proof.
I usually experience lag with jam sessions, but unless you're telling me everyone that participated experienced it (in which case, that would've just been really bad luck), it should only be on your side.
Maybe I should've just uploaded my 3MLE file itself onto Mediafire. Too late for that since my computer's dead.
You have rank 1 composing.. right..?