I can answer part of your question in reguards of HostMonster and BlueHost are both certified Zen Cart hosting companies directly from the Zen Cart development team.
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<mgraphic /> - I don't have a solution but I admire the problem.
Does Hostgator have "private domain name registration" so that my contact info is not publicly available on the registar?
While I am and have been satisfied with their hosting service for years, I still keep my domains with a third party company. It's always the smart thing to do, keeping the domains separate from the hosts you use. I use Namecheap to register my domains, and they offer 1 year of whois protection with any new domain or domain transfer.
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That is called whois protection. But be sure that your information will be accessible for the one who need it.
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I don’t see any reason why you need to have your domain registered by different company that is of course if you don’t use use of those WHC <link drop removed>