No. Don't use Notepad++.I would suggest getting Notepad++. It's like MS Notepad but sooooo much better. It's very useful for quickly going back and forth between code.
And you technically don't even need MS Visual Studio. Tmodloader works kind of like scripting. There really isn't compiling in my opinion.
Mods can do pretty much anything C# can do. Also, you can, in fact, use external libraries.I also just started to learn how to mod recently and I get most of the information that I from the official Github documentation.
1. I believe that the only limit to modding is that you can't do anything that the base game can't do. Plus I don't think it's possible to use any external libraries (dll files for example).
2. As stated in the documentation, Visual Studio is probably the best option available.
3. Tmodloader has an inbuild publishing option :
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Yes. It's also the IDE that most of the tMod API devs use.The most major reason to use Visual studio is Intellisense and all the useful plugins/addons