does the eval mode of ESX also expire after 60 days?
Yes, same eval period after installation.
(if above is yes) can I switch my ESX hosts back to the free key that doesn't expire when the 60 days is up?
Yes, you can switch licenses any time. However the free license is not supported in a vCenter Server configuration. (Even though this may work during the eval period of vCenter Server)
what is the purpose of the free ESX key then?
Saving money. If you only need a single host (or multiple single hosts) without the need to be managed by vCenter, you can use the free license.
André