What exactly could be accomplished with a backdoor in Dual_EC_DRBG? Assume that some entity really holds the private key corresponding to the recommended/dubious constants of Dual_EC_DRBG.

According to this presentation, they would be able to reconstruct the internal state from only 32 bytes of random output and thereby predict all future output as well.

Would the backdoor also allow recovering previous output of the RNG, or does the 
resistance against state compromise extensions still hold in that situation?
State update takes $s$ to $x(sP)$, so you would have to compute d.log.s to run the state backwards.