Can I show that the "random user" feature of a website is fair? I have a script selecting random users from deviantArt using their "Random user" feature. 
Is there a way to be confident that the users that they select are unbiased--that every user has an equal chance of being selected for each query?
There are about 35 million users according to the website and I've (so far) retrieved a total of 2000 random users over the past few weeks so far. I heavily throttle my access so that my normal use as a user is far heavier than the bandwidth that the script uses. Their TOS seems to be ok with it also.
Can I analyze my data to be confident that the samples are fair? I have many quantitative properties that I record for each user. Perhaps those could give me clues about the fairness?
Would it work to select several random (say) 90% subsets of my samples and compare their mean and standard deviation on each of my quantitative measures? If the subsets are consistent on each measure with the whole sample, can I be confident enough? 
What topics and terms should I study so that I can explain my situation correctly in a formal paper?
[I might be misusing statistics terminology. It's been a long time since I've studied it.]
 A: 
Can I analyze my data to be confident that the samples are fair?

No. You can check for particular kinds of unfairness, but failure to detect unfairness doesn't mean that it's perfectly fair, even in exactly the ways you checked. The best you might hope to demonstrate is that it's not more than a certain amount of unfair.
You can test various hypotheses but failure to reject a null of fairness doesn't mean it is fair.
You might be best served by considering equivalence/noninferiority tests - to identify what you might regard as "practically fair"* and then test for that in dimensions you deem relevant
* consider if you had two artist types, A and B, and it turns out that random individuals from A and B don't have identical probability of being selected -- artists who are A's are selected slightly more often than artists who are B's, relative to their prevalence. This is unfair but if that advantage was small enough it may not really matter (if the guy down the road has a 1% better chance than me is that something I should expend energy to worry about?). 
You may be able to bound the unfairness and say "well, it's not worse than that".

I have many quantitative properties that I record for each user. Perhaps those could give me clues about the fairness?

If you think there might be bias related to those particular properties, perhaps (e.g. if they're potentially discriminating by portfolio size or genre or something)

Would it work to select several random (say) 90% subsets of my samples and compare their mean and standard deviation on each of my quantitative measures? If the subsets are consistent on each measure with the whole sample, can I be confident enough?

I don't think this achieves what you want. 90% subsets of your own sample will reflect the information already in your sample. If such a scheme would work you
could get similar information from the whole. Resampling your sample without replacement (bootstrapping) is sometimes used to measure variability of estimates
but it doesn't quite work the way you outline. I also don't think it's necessary 
when modelling a proportion in this way, but there can be situations where it might be useful.

What topics and terms should I study so that I can explain my situation correctly in a formal paper?

I think this is too vague/broad as it stands.
