Answer is you don’t have to initialize rcx, since it is already a prerequisite for a looping subroutine. There is no for (i=0; i<n; i++). Argument n is a counter and it is rcx itself. It is ‘discharged’ by a loop. If caller wants to save it, then it does that by his own means. This contradicts all regular languages’ rules, but this is low-level, far below all the safety.
On your second question about different loops. First, you can loop through any register via (dec <reg>; jnz <label>), thus having nested loops. Second, all valuable registers are pushed onto the stack and popped before/after a call or an interrupt, so their contents don’t change. Some registers have to be saved by the caller and some by the callee, depending on a calling convention and an actual usage. The stack is always available (rsp points to its top), so you can offload temporary values and concentrate on current evaluation, loading values on demand later.
On your second question about different loops. First, you can loop through any register via (dec <reg>; jnz <label>), thus having nested loops. Second, all valuable registers are pushed onto the stack and popped before/after a call or an interrupt, so their contents don’t change. Some registers have to be saved by the caller and some by the callee, depending on a calling convention and an actual usage. The stack is always available (rsp points to its top), so you can offload temporary values and concentrate on current evaluation, loading values on demand later.