Can C# apps run without the .NET framework?

No. c# only target .NET (or a comparable framework, such as mono). As an aside, Win7 comes with .NET preinstalled, and I believe Vista did as well. There are also a ton of MS apps which require .NET. It's getting near ubiquitous on windows machines, so I wouldn't worry about it.

YES, there was XenoCode that can wrap everything that your app needs and runs it in as a standalone. I don't know what kind of dirty tricks they use, but there IS a way.

Now it's Spoon

From their site:

Spoon Studio

Easily virtualize all of your applications for instant, zero-install delivery on Spoon Server and Spoon.net.

Spoon Studio lets you convert your existing software applications into virtual applications that run with no installs, conflicts, or dependencies.

BTW, I'm in no way affiliated with them - just curious if the community will accept it or flame it.

Yes, with .NET Native.

Instead of compiling to intermediate language, it will compile to native code and run with statically linked .NET libraries. Therefore, there will be no .NET Runtime requirements for end users.

https://msdn.microsoft.com/en-us/vstudio/dn642499.aspx

https://msdn.microsoft.com/en-us/library/dn584397(v=vs.110).aspx

Only works for Windows 10