In Robo Rally, does a powered down robot get to earn option cards and repair?
It depends on how you view the rules from the original 1994 edition vs the rules from the newer edition.

In the rules for the original edition, you cannot:


  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. 


This is also clarified in the FAQ from that edition:


  Q: What can a robot do when it is powered down?
  
  
    A: Nothing. A powered down robot cannot fire weapons, receive option cards, tag checkpoints (if pushed onto them), update archive locations (ditto), or use turn programmed cards (for example, Shield). It cannot use any options unless the options specifically say they can be used while a robot is powered down. The robot basically becomes a lump that can be pushed around and suffer damage.
  


It is reasonable to assume that rules from the original edition are intended to be carried to the new edition unless it's somethign that the new edition specifies (such as the differences in how robots re-enter the board). Based on this, the answer is no, the intent of the rules is that you cannot receive new option cards or repair while powered down.

However, this rule is missing from the newer edition, so by a literal reading of the newest rulebook, without being concerned about the old edition, the answer would be yes. The rules for earning new options simply say that a robot on the correct space earns an option, no exceptions listed for if that robot is powered down. And the rules under powering down make no mention at all about this preventing you from earning new options or healing. The rules do specify that your robot can still be affected by board elements, and lists taking damage and conveyor belts as examples of this. One could say that option and healing squares fall in this category as well.

There's some good discussion on this question here and here, but no official answer given.