These are valid questions a lot of people new to Rust have, so:
1. Rust is "backward compatible" in the sense that Rust code can use C libraries and C code can use Rust libraries - both ways via CFFI [1]. Security gaurentees only apply to the Rust code.
2. We've tired static and dynamic analysis of C to find security bugs for decades, there has been a plethora of research and commercial tools in the space. None fix the problem like Rust does [2].
Almost any language can call C functions and we don't call all languages backwards compatible with C when they can merely interoperate with it.
Objective-C and C++ are the only two languages which offer backwards compatibility. AFAIK it's complete in the case of the former and there are some limitations for the latter.
None fix the problem like Rust does, but it's worthwhile to examine why: typical companies and developers have an aversion to paying for tools and for anything which slows down development. That's why usually those tools and languages which are reasonably user-friendly are more successful. Ironically that's both an advantage and a problem for rust: it's nicer to use than some C tools, but still not user-friendly compared to alternatives like Go or Java and in some cases even C++.
Objective-C and C++ are the only two languages which offer backwards compatibility. AFAIK it's complete in the case of the former and there are some limitations for the latter.
None fix the problem like Rust does, but it's worthwhile to examine why: typical companies and developers have an aversion to paying for tools and for anything which slows down development. That's why usually those tools and languages which are reasonably user-friendly are more successful. Ironically that's both an advantage and a problem for rust: it's nicer to use than some C tools, but still not user-friendly compared to alternatives like Go or Java and in some cases even C++.