This is exactly how Firefox is approaching it. Identify the most security-critical bits, wall them off into their own component, and rewrite it in Rust.
Yes! And the reverse has happened with Servo. Originally it was mainly a bag of C libraries with some layout code, then gradually they began swapping the pieces out for the Rust counterparts. This is how new software can be written - bootstrapping off the C infrastructure, then gradually transitioning more and more into Rust land.