No. That kind of restriction cannot realistically be applied to any project above toy scale. The stdlib uses unsafe code to implement a large number of memory management primitives, because the language is (by design!) not complex enough to express every necessary feature in just safe code. Rust's intention is merely to limit the amount of unsafe code as much as possible.
No, and in fact that would be impractical, because you can't do anything useful (e.g., any I/O whatsoever) without ultimately either calling into a non-Rust library or issuing system calls directly, both of which are unsafe.