Crossposting from MathOverflow is perfectly fine, as long as they aren't done in parallel. That is, if you post a question on one site, you should only post to the other site after you have not received a satisfactory answer for some time, and you should provide a link in each post (or in the comments on it) to the other one. As a courtesy, if you post your question here after trying MathOverflow, please try to integrate in your question the answers you received on MathOverflow (even if they did not answer everything).

At some point I was proposing that we remove the 'in parallel' restriction, but consensus was that we should keep it in place, but enforce gently.

D. Is there anything we can do to dispel the perception that MO is a "last-resorts" "ask-an-expert" place? I rather selfishly would like MO to stay a forum for research mathematics, and not a forum where "professional mathematicians hang-out and answer questions for which answers were not found elsewhere" becomes the norm.

Perhaps the suggestions you list above w.r.t. C should also be included in the proposed MO faq change? That is, something along the line that

1. Rather than the shot-gun approach of shooting questions simultaneously to MO, MSE, and CStheory, we encourage a more surgical approach of asking the question in the most appropriate forum.
2. If one must cross-post, please have the decency to provide links to the cross-posts
3. Perhaps a suggestion for the poster to exercise the power to "close as no-longer relevant" if an answer was received on other sites? That one cannot easily coordinate answer given on different fora is one of the reasons I find cross-posting undesirable.

A. Is cross-posting really a problem on MO?

This is unclear to me. I spend a lot of time monitoring annoyances on MO, so far the annoyances associated with cross-posting have relatively low frequency.

B. What, if anything, should be done about cross-posting?

Even if cross-posting is not an emergency problem, we can still do something about it. A brief addition to the MO FAQ can't hurt, though it should be carefully worded.

C. What is the appropriate way to handle cross-posts?

There are several issues here.

Ideally, the cross-poster would diligently link to the cross-posted question and its answers to avoid redundancy and confusion. Since users aren't that reliable, I would suggest adding missing links when anyone notices a cross-posted question.

What if a cross-posted question gets an accepted answer on another site... Should the question be closed (no longer relevant)? Should it remain open for new answers? Should the accepted answer be cross-posted as well?

@Andrey: I think that is too extreme. Re-posting is warranted at times (there was a question which was on Math.SE for 2 or 3 days at least before it got re-posted here [which I was about to do myself if the owner didn't] and received a great answer in 2 or 3 hours). But perhaps direct cross-posting (MO, Math.SE, cstheory) should be forbidden?

@Andres: I agree. What if I add something in the FAQ of Math.SE to the effect of "be patient, just because your question is not yet answered [in 5,6, 10 hours] doesn't mean that it won't be; many users live on a different time zone from you"?

Some users seems to think that just because a question is asked on MSE and did not receive the answer that they want, the question should be automatically upgraded to MO. In some cases this is warranted: the user may have hit upon a question that is of reasonable research interest, or one that is sufficiently thought out and clearly written that a real expert can easily spend five seconds of his time giving an answer. In many other cases, this is not: often part of the reaons that the questions did not get an answer on MSE is because that they are not sufficiently well posed mathematically, and in some cases the questions have limited research interest.

To me, this feels like an abuse of the MO mandate, as stated in the FAQ (which unfortunately nobody reads).

Now, my questions/comments:

1. How can we properly discourage direct cross-posting?
2. How to respond when a user carries a question that doesn't quite fit the MO scope, but that he/she already asked on MSE and received no answer for? (In particular, the old stand-by "please try asking this on MSE" would be rather inappropriate in this case.)
3. Related to (2) is I think again an old discussion rearing its ugly head: for us mathematicians we know a research-level question when we see one (some people are more generous than others, so there is a fuzzy belt in the middle, but for large parts I think people agree on what research-level roughly means). But those users whose behaviour this post is trying to affect are precisely those for whom the phrase "research level" just means "difficult". So how do we properly convey the notion that a good question for MO should be one that is well-posed mathematically, and ideally also with some motivation?

(Tangentially related to (3) is a recent question on MSE.)