Clearly I’m not thinking big enough!
If I understand you correctly, parts of the function could arise on different parts of the genome? Also at different times, or the same time, would seem to follow. I think we might even see multiple functions from some units, but I’ll leave that alone.
I didn’t know there was going to be homework! 
how long before 1 safe is opened?
For one specific safe this looks like a Geometric distribution, clearly longer codes will be less likely (unless there is repetition? ignoring that).
Wild guess: probability \propto1- {bits \over thieves }
No … that can’t be right …
For multiple safes, much faster, as I describe in Multiplying Probabilities.
How long before 10 are opened?
Given probability p of finding some function, this is a Negative Binomial probability distribution.
How long before x are opened if x is much less than the total number of safes and the key size is non trivially long?
Also negative binomial, or approximately so. With each discovered function the probability of the next will be slightly less, assuming a finite number of functions.
How does it scale with the number of thieves (genome size)?
Depends on which “IT” you mean. But bigger genomes means many more opportunities to discover function, similar to my Multiplying Probabilities situation.
I think I’m failing this homework, but I haven’t Googled. 