Can a powered-down robot be repaired at a workshop?
Yes, robots can still receive damage tokens while powered down. From Page 5 of the rulebook under Power Down:


  Other robots can push a powered-down robot, and the robot can still be damaged (such as by laser fire), Because of that, a robot that’s powered down may receive new Damage tokens during the turn


Yes, you still repair damage and collect option cards. (Page 5)


  The robot doesn’t receive or execute Program cards and doesn’t move while powered down, but board elements still affect it.


And page 8:


  Repairs &amp; Upgrades - Robots on a single-wrench space discard 1 Damage token. Robots on a crossed wrench/hammer space discard 1 Damage token AND draw one Option card. 


Note: My answer is based upon the most recent rules available from Wizards of the Coast. There seems to be some differences between older printings of this game. This thread claims to link to a text version of the 1994 rules that indicate:


  A powered-down robot is completely shut down--it can't fire weapons, tag checkpoints, or update its archive location, nor can it acquire or use option cards. Powered-down robots don't move under their own power (they receive no program cards), but they may be moved by pushers, gears, and conveyor belts. They may also be pushed by other robots and shot at by lasers