Your 10:57 retime appears to be off by about 15 seconds. Of course, that doesn't matter, since the time is not going to get retroactively changed anyway.
No. From the RSG co-op rules:
"You must use a randomly generated seed OR Geosquare's Random Seed Glitchless Co-op Spawn Finder which you can find in Resources (ONLY ALLOWED FOR PRE 1.9)."
Texture packs are allowed if they don't give you a significant advantage. Fullbright is not allowed; the maximum allowed gamma is 5. Coordinates are allowed, but make sure to mark your run as "F3".
The in-game timer is a standalone program; you can't put it into Livesplit.
It could also be the unavoidable lag from when the game auto-saves every 5 minutes. If that's what you're experiencing, you'll just have to deal with it.
Directly from the rules: "Badlion client is considered a mod and is banned."