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If you just straight up loaded in files without clearing out the old ones in /mods/, then you have duplicates of everything that is updated.
Some ones I can note just from your log:
DartCraft
Forge MultiPart
Logistics Pipes
Natura
Tinkers' Construct
MiscPeripherals
If you clear out /mods/ and then upload the /mods/ included in the server .zip, you should be okay. Make sure to also do this with /config/. As always, back things up before proceeding.