Does Stack Exchange or MathOverflow own my posts? Can I reuse my posts elsewhere? Do I need to cite MathOverflow if I want to reuse my posts elsewhere?
You own all of your posts! You retain full copyrights and you can do anything you please with them.
However, by posting on MathOverflow, you release your content under the terms of the Creative Commons Attribution-ShareAlike 3.0 (CC BY-SA 3.0) license. This license allows MathOverflow, Stack Exchange and others to use your content under specific terms. This is not an exclusive license. As the copyright owner, you may release your content to anyone under any terms you please, though such waivers do not revoke the CC BY-SA 3.0 license that you previously agreed to.
EDIT: what is written in this answer does not apply anymore as the TOS have been changed.
EDIT: as @FrançoisG.Dorais and @ScottMorrison point out in the comment, my interpretation here below is probably wrong, too. What is true is that, with the current TOS, if you re-use your own content elsewhere, you have to link to MO as the original source of it.
@FrançoisG.Dorais 's answer is incorrect, at least under a reasonable interpretation of the Stack Exchange terms of service; see here for more detail.
Apparently, by posting content, you restrict everybody (including yourself) to use the contributions only under the CC BY-SA license.
In particular, you cannot use that material in a paper or a book, unless it's licensed as CC BY-SA itself. Given the current state of affairs in academic publishing, good luck getting your publisher to agree to that.
At the same time, Stack Exchange and everyone else in the world has the same rights, so after you post on MO you don't "own" your writing more than Stack Exchange, or I, or anyone else in the world does.