In pure Rust that never uses "unsafe", maybe. (Though even then, you only get memory safety; it's still possible to e.g. go into an infinite loop and use all CPU and/or memory).
Rust intercepts most stack overflows and aborts the program when they occur. There are some more difficult cases where legit stack overflows are still possible, the resolution to which is blocked on LLVM support on most platforms (notably, Windows is the best here).
Well ultimately any useful program will use a library that uses unsafe. All I/O is unsafe at the bare metal so while you can safely wrap it it still is unsafe internallly.
When talking about the usage of unsafe code in rust, we generally ignore encapsulated unsafe code in libstd since it has the same safety guarantees as tgr compiler itself. all unsafe code encapsulated in a safe API has these guarantees, but you trust libstd more :)
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).
