First is what exactly is the "core" of Lisp? What is the bare minimum from which other things can be re-created, if needed?
Most Lisps have a core of primitive constructs, which is usually written in C (or maybe assembly). The usual reason for choosing those languages is performance. The bare minimum from which other things can be re-created depends on how bare-minimum you want to go. That is to say, you don't need much to be Turing-complete. You really only need lambdas for your language to have a bare minimum, from which other things can be created. Though, typically, people also include defmacro, cond, defun, etc. Those things aren't strictly necessary, but are probably what you mean by "bare minimum" and what people usually include as primitive language constructs.
The second part is where can one look at the code of the macros which come as part of Common Lisp, but were actually written in Lisp?
Typically, you look in in the Lisp sources of the language. Sometimes, though, your macro is not a genuine macro and is a primitive language construct. For such things, you may also need to look in the C sources to see how these really primitive things are implemented.
Of course, if your Lisp implementation is not open-source, you need to disassemble its binary files and look at them piece-by-piece in order to understand how primitives are implemented.
As a side question, when one writes a Lisp implementation, in what language does he do it?
As I said above, C is a common choice, and assembly used to be more common. Though, there are Lisps written in high-level languages like Ruby, Python, Haskell, and even Lisp itself. The trade-off here is performance vs. readability and comprehensibility.
If you want a more-or-less canonical example of a Lisp to look at which is totally open-source, check out Emacs' source code. Of course, this isn't Common Lisp, although there is a cl package in the Emacs core which implements a fairly large subset of Common Lisp.
