How does multi-core CPU implement asynchronous coordination?
You mix two independent (orthogonal) ideas in digital circuits theory: asynchronous circuits and multi-core processors.

Asynchronous circuits: circuits which have more than one clock, and the clocks are asynchronous (i.e. have non-constant and unpredictable phase relationship). 

Some circuits may use two clocks (for example), but one is just a division by 2 of the other. These circuits are not asynchronous because there is known phase relationship between the two clocks, although the frequencies of the clocks are different.

You may have a single core CPU having few asynchronous clocks, and a multi-core CPU with all its cores running on the same clock (the latter is just an imaginary CPU - all real multi-core CPUs have many clocks which consist several mutually-asynchronous clock sets).

Asynchronous circuits is a major topic in digital design. The above explanation is basic.

Multi-core CPUs: few microprocessors (cores) connected in parallel which employ sophisticated hardware and software in order to achieve high performance. 

The usual practice is to make the cores as independent as possible in terms of clocks/power/execution/etc. This allows dynamic (at run time) adjustment of CPUs activity (i.e. consumed power) to the actual needs of the system.

My impression is that what you're looking for is an explanation about multi-core CPUs, not asynchronous circuits.

This topic is much, much bigger than anything one can put in the answer.

The answers to your questions, though:


The clocks used by different cores (to my best knowledge) have the same sources (can be more than one: crystal, VCO, ...). Each core (usually) has few mutually-asynchronous clock sets. Each core has dedicated clock gating and throttling logic which allow to turn-off or slow the clock, independently for each core. Again, if you're interested only in algorithmic aspect of cores' parallelism - forget about clocks (for now).
You have just indicated the main aspect of cores' parallelism - how do you run multiple cores in parallel efficiently. This topic is huge, and contains both HW and SW solutions. From HW perspective, cores both modify a common memory and exchange control and status signals with sequencing logic and between themselves. The picture complicates a lot due to existence of caches - I'd suggest that you start from reading on caches, then cache coherency, and only then on cashes in multi-cores systems.


Hope this helps.