Even in the heat of summer, my rotary cutter gearbox only gets warm to the touch, not what I'd call hot. It was shipped with a plain plug in the gearbox, and a vented one in an attached bag to replace the solid one once it was in service.
The anaerobic sealers such as Loctite 515/518 etc won't harden until there's no air in contact with the sealant, so simply tighten them to the snug point and wait 24 hours before adding oil for the best results. A final tighten won't hurt matters any, and they will work if you add oil right away too, but it's not ideal.
I've only ever used plain old 80w90 gear oil and had no problems, but there's no harm in using the high end synthetics if it makes you feel better.
Sean