Scanning over the questions asked...
My guild rep cap isn't resetting every Tuesday like is should, what can I do?
There was a hotfix on the 27th that hopefully addressed this problem:
# Weekly guild reputation caps should now properly reset at the scheduled time. An error which prevented this reset from taking place if guild experience was earned at exactly 3:00 AM server time immediately prior to weekly maintenance has been corrected.
If you still experience problems with it next week I'd submit another GM ticket.
How else do I accumulate Guild Reputation other than questing?
The activities that earn you guild reputation are the same activities that earn you guild experience. Turning in Quests (regular or daily), Guild Group Boss Kills, and Guild Group pvp activities (may be rated battlegrounds only?)
Can I earn Guild Achievement credit for completing a lower level instance if I solo it myself or complete it with a group less than 4 players but comprised entirely of guild members?
In theory the answer is no, though I haven't tested that.