I will give some brief answers because the mailing list is not a place for general simulation advice but more so a place for utilization of the LAMMPS software.
(1) As a bit of reassurance, the fix deform command and periodic boundary implementation has been tested countless times. It is highly unlikely that the issue (if there is one) is on your end and not with LAMMPS.
(2) One polymer chain can cross the same periodic boundary multiple times if it is coiled around it. Deforming this chain could cause it to elongate and no longer cross the boundary multiple times; hence you could “lose periodic bonds” but still have the same number of load bearing chains in your network.
(3) Bad geometry? Are you seeing anomalous temperature or potential energy jumps? Generally these phenomena always accompany bad geometry. Wouldn’t this mean that the higher rate would have issues too? They have the same initial geometry…
(4) It seems like you have a good idea of the macroscopic picture of your network but not the microscopic simulation picture; I HIGHLY recommend that you compare the total number of bonded terms in your system (to see if you are actually losing anything) and generate bond/angle histograms / radial distribution functions to understand the differences in your systems.
I hope this helps. I am confident, based off what I have seen, that this is an issue with your understanding of simulating polymer networks and not with LAMMPS or utilizing the LAMMPS commands. Sounds like LIT REVIEW to me!