Surely if one programming language can do it, another can?
A C compiler that checks all of these things at compile time would no longer be a C compiler.
Compile time guarantees require you to change the language and restrict what is allowed to compile. Rust does just that, it's a different language. You could write an extension to C like Cyclone (or the ISOCPPCore guidelines for C++) that make it safer via compile time checks. You would likely need more annotations and most existing C programs would no longer compile.
(There is the ergonomic benefit of being able to transition from a C codebase to a, say, Cyclone one, though)
No, due to the way the language semantics works.
There are static analyzers that do something like that, MISRA-C, High Integrity C, Frama-C, but you are literally using C with Ada semantics at that point, thus it is almost like another language.
From my experience though, Rust is the first language that I've had the pleasure to use which offers both the safety guarantees of Java (and more, no data races!) and the low level features of C.
It still leaves me in awe what has been accomplished with Rust, and I've been using it for more than two years now.