As much as I think C# at a platform level is a better tool for building backends, you'll get the better bang for the buck learning TypeScript if you don't already know TypeScript.

Then if you have the chance, you'll find C# an easy transition from TypeScript, IME. Learning C# first, on the other hand, will make you a better TS developer, in my opinion, because it will shape your approach to be more diligent about using types. This is something most JS/TS devs do very poorly and at scale, it's very hard to reason about code when it requires digging down several layers to find the actual types/shapes.

"Enterprise" frameworks like Nest.js are much more similar to ASP.NET or Spring Boot than they are to Express, Hono, or Elysia so once having experience with .NET Web APIs (or Spring Boot) will make Nest.js (for example) easier to pick up.

Not really, you should learn Typescript by learning JavaScript first. Then consider learning C#. Or if you want to focus on the back end side learn C# and skip TS/JS.

They are created by the same person but they are very different in my opinion.

TypeScript is "a tool" for JS, it is possible to compile without errors but still fail in runtime (e.g. wrong object type returned from API), on the other hand parsing JSON with C# will give you correct object type, it may fail if some properties are missing but it will fail at parsing call, not further down when you try to use missing property. In other words typing is not glued on top of the language it's core of the language.