I am not really sure if you mean ASP.NET, or really a .NET Desktop or even mobile Application, but I assume you mean the latter ones, as you already will have a clue how to insert a Leaflet Map into a ASP.NET application.
If you don't have a commercial framework like ESRI's ArcEngine or mapdotnet.com available, one possible option is SharpMap. There are much more for sure, but I would take a look at these three as I already know them.
The SharpMap Tutorial Page covers everything you mention, so it should do it.
P.S. I am not valueing any of these frameworks. Each one has its pros and cons and it depends on requirements and environments which one is the best to choose.
Another option would be MapSurfer.NET framework. This is a rather advanced tool to solve different cartographic tasks. It is based on some OGC standards, for example, it supports WMS and TMS specifications. The installer has SDK package with a sample project (Visual Studio) for a desktop application, where you can visualize your own map in a control that allows zooming and panning. Unfortunately, this framework lacks any documentation. But, if you are familiar with GeoServer, MapServer or Mapnik, it won't be that difficult to grasp how it works. You can find a couple of maps generated with the help of MapSurfer.NET on OpenMapSurfer page.
You may try AspMap ASP.NET-based component, although it's a commercial one.