I only had an intro course in CPU architecture, but AFAIK pipelining depends on parts of the pipeline running on the same clock. I'd say CPU manufacturers already sidestepped this limitation by using multicore dies.
That just means you can't have a single simple clock source fan out in a simple way across the entire chip, but that kind of design performs badly anyway.
There are lots of ways to have multiple clock sources kept in sync or to alter the clock's phase to compensate for distance, so you can keep the entire CPU synchronized even if crossing it takes multiple clock cycles.
No, it doesn’t. Putting aside that there is an entire field of clockless designs, asynchronous logic, free-running clocks, etc.
You should think of this as “when the cpu is processing an instruction, what does it need to track to dispatch and retire it?” Designing to a click tick is one version and kind of where things were in the 1970s. It’s an educational model.
