Suppose I create some software from scratch. I sell the software, but I also keep it open source under the MIT License. Someone forks my repository and makes some great improvements to my software. Can I then clone their repository, and sell the improved software without sharing any profits?
Depending on how they chose to provide their fork, yes.
The MIT license, which you chose to license your work under, doesn't prevent anyone downstream from changing the license, nor from changing the license of a derivative - unlike a copyleft license such as the GPL. They can essentially license their fork how they like.
If they relicense it under CC BY-NC, you're not allowed to sell it because of the NonCommercial clause in BY-NC. However, that's not a free license.
If they've relicensed it under a free license (of which there's a list on the FSF website), then by definition this allows you to redistribute their work and sell it on.
Keep in mind, however, that if they're providing the same package for free, you're not going to get many customers simply because it's cheaper elsewhere.
Yes, provided they license their improvements under MIT or another free/libre license.
Please note that MIT (Expat) is a permissive license, so the fork may no longer be free/libre software - the improvements may even be proprietary (ARR) - in that case, the other party will be able to sell the improved version, but you will not have that option (but you can still sell the your original version).
If you use a Copyleft license (e.g. GPLv3) instead of MIT/Expat, they can't do this. Then any improvements to your program must be licensed under the same license.
However, if the improvements have a free/libre license, you're allowed to sell free software, even if you don't hold the full copyright (or any copyright).
Also see: http://www.gnu.org/philosophy/selling.en.html (while this is from the FSF, it also applies to permissive licenses such as MIT (Expat)).