Free tool for exploring compiled .NET assembles, alternative to .NET Reflector?

ILSpy is one of the more popular FOSS .NET decompilers. It's pretty intuitive to use, though I have not yet tried Reflector so I can't compare them.

Unfortunately, it doesn't completely satisfy your requirements, though it gets close. There is no support for decompiling to C++/CLI, though it does work well for IL, C# and VB.NET.

You can click on class and method names to navigate to their definitions, or analyse them through the context menu for where they are accessed. It can render both images and text from embedded resources, and allows you to save them - I have not tried other filetypes, but they should work in the same way.

You can export classes to source files, and you can even export entire assemblies into .csproj (MS build) files. For simpler projects these can be compiled directly. Many more complex projects tend to fail to compile without modification - notably, their site mentions it can't decompile and then recompile itself. If you need to decompile an entire assembly, you would likely have a lot of work left to get it to a compilable state.

Unfortunately, decompilation is not perfect. Some more complex language constructs, like C#'s dynamic, are not yet supported and can result in somewhat garbled output. You can still get the general algorithm through the clutter, but it certainly wouldn't work if you just wanted to copy some code.

IDE integration. There was an early attempt for Visual Studio, though it appears to have been abandoned. There also doesn't seem to be any integration for Xamarin (when'd that happen?). However, the developers of ILSpy also created SharpDevelop, and recent news indicates they are planning SharpDevelop integration.

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If you want other options, there's also dotPeek and JustDecompile. They do claim to have Visual Studio extensions, and might be more polished with some well-known companies behind them. I have not had a need to use either of them (ILSpy has worked for everything I needed), but they may suit your requirements better.