Is there a computer which can simulate all computers?
Your intuition sounds wrong to me.  It's not surprising you couldn't prove it, because it isn't true.  Roughly speaking, it is possible to find a computer that can simulate all other computers.

A universal Turing machine can simulate any other Turing machine, and can be used to simulate any other program in any Turing-complete language.  Thus, there is a reasonable sense in which it can simulate all other computers, if we make precise what we mean by "computers" in a particular way (e.g., if we accept the Church-Turing thesis).

If you want to know whether it is possible to simulate another computer and also do it faster, that's a different matter.  You'd have to formalize what you mean by "faster" to make that question well-defined.

Your question at the end of the post is not well-formed; computability is a property of languages, not of a single value (like $f(f,i_f)$).  Also, you defined $f$ to be a function of two arguments, so $f(i_f)$ doesn't type-check.