Arch Linux. Here's a quick brown townysis of those Linux distros from my experience:
Mint: Ubuntu with a twist. Useless for servers, you may as well just use Ubuntu because this doesn't add anything special.
CentOS: Terribly stuffty package manager, everything is ridiculously complicated to set up.
Fedora: Slow, old. I honestly can't even think of one server that runs it.
Debian: My personal favorite of the above, it's a little bit harder to use than ubuntu but it's less bogged down by stuff you don't even know you have.
Ubuntu: The worst linux distro to ever appear. In their attempt to make linux user friendly, they forgot that 99.99% of the people who use linux actually know what a computer is and now it's incredibly difficult to use. May be easier through command line, but it also comes with like 500gb of packages you didn't even know existed and don't know what they're used for, but if you try to uninstall one it's a dependency of 50 other packages you didn't know existed.
Gentoo: People still use this? I've never used it personally, so I can't really say anything about it.
If it's possible, go with Arch Linux.
Arch Linux: Comes bare bones. Install only the packages you need, one command makes this an absolute breeze. It's called Yaourt, and it will install and configure virtually every program that can run on linux flawlessly. This distro has the most comprehensive wiki of all linux distros, and their forum is almost a personal tech team to help you solve your problems that may not be covered by the wiki articles.
All of these distros will run wine.
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