What is a "more dedicated language" in this respect? Your options are C, C++, or Rust. Same usual comparisons apply, I don't think wasm is particularly different enough of a situation to have any special advantages.
Not in the world. The options depend on a use case. For a website C++ and rust would be far down on the list. For a web server, even, there is a plenty of competition.
Similarly for an embedded scripting language or rapid prototyping.
I'd argue that safety is vital for web software, so Rust is a candidate for "the best language" for web servers. The same is true for security libraries.
Or how about the original purpose of Rust, to rewrite a rendering engine? That's an incredibly widely used platform.
There's an OS being written in Rust that a lot of people are really excited about. There are very few languages other than C/C++ that are actually viable for rewriting an OS and would, in and of themselves, prevent entire classes of bugs.
Similarly for an embedded scripting language or rapid prototyping.