This is not true. If you know Rust and C equally well, you're likely going to write equally performant Rust.

You could say that Rust is harder to learn than C. I'd disagree based on my personal experience, but you wouldn't be wrong.

Well, you need to type more and you need to learn more things with Rust, before you can start making stuff.
But the additional work is to make it easier for you to make changes later, when you come back to it after a while.

So you might need to do more before hello world, but say if you have a complex library and want to use some function of it after learning Rust, it will be easier to not make some common mistakes.

A pretty good recent example of something that will cause a common mistake would be:
In the mongoc library, there is a function named mongoc_client_select_server and the pointer it returns requires destruction using mongoc_server_description_destroy. But it doesn't say so in the function's comments/documentation. So, I had to go into the function called by the function called by the function called by it, to find the function making said pointer and having a comment stating that the pointer made by it would require destruction by the user.
And the only reason I found that out was my obsession, but I had already made the mistake.