Any reason C# and Mono + MonoGame is not allowed?
Can we use DirectX, assuming the program compiles on MINGW and runs on WINE? What about Mono?
We strongly recommend against it, as we can't guarantee that entreies that use DirectX or Mono will be acceptable under the rules of the contest. It may be theoretically possible to use the DirectX API in compliance with the contest rules, provided the program compiles with MINGW and doesn't require any proprietary header files to do so, then runs on an out-of-the-box WINE install that does not include any proprietary Microsoft DLLs.
Secondly, questions about whether Microsoft or a third party might make patent claims about Mono have never been completely resolved (Microsoft's "promise" not to sue for patent infringement is not a legally binding agreement). It's not a matter of debate that Mono is free software, but considering the fact that Microsoft has quite pointedly refused to make a legally binding promise not to sue, you should understand that using Mono does to some extent go against the spirit of the Liberated Pixel Cup, and may negatively affect your score.
TL;DR: Maybe, but we strongly recommend against it.