Yes, quite a bit of it: https://github.com/servo/servo/search?l=rust&q=unsafe

Any reasonably large program will have at least some amounts of unsafe code. And that's not a big deal. The point of Rust is to very clearly isolate the unsafe parts, so that you only need to audit 1% or less for memory safety.

If it uses any external libraries (for example, for image handling) then I believe it does by definition, since the API wrapper you end up writing ends up being a small shim that wraps the actual library call in unsafe (because object code is unsafe, as the term in used in Rust).