I assume one repairs 10% on an occupied system, less elsewhere. Do nanorepair bots only work on the ship they are on or anywhere? And does the benefit stack? Do you get better repair as system levels rise? Any other good ways to get one's fleets back in action?

The base repair/turn ratio is indeed 10%, although Unfallen got a bit more (15% I think) and it can be increased in several ways. The one available for any faction is to research Advanced Citadel Upgrade and build Advanced System Battery. That will increase regen to 20% when docked. Also assimilating Sisters of Mercy adds to that value. Cravers have improved regen of +20% as one of the quest rewards.

Thanks - I still haven't played long enough to explore into the higher reaches of the tech trees!