>> "When I release a new version of my software, will LimeLM automatically create a new product key for my existing users?"
Automatically in what sense? If you have some soft of automated payment method like here, then the script will continue to generate product keys for whatever product / version you set it to generate keys for: https://wyday.com/limelm/help/how-to-generate-product-keys-after-order/
>> "I'm assuming a new product key will be needed, since a new LimeLM version has a totally separate TA.dat file and Version GUID."
Yes, if you create a new product version in LimeLM the product keys are not compatible with any other product version created in LimeLM.
>> "What do I need to do to make my users' transition to a new version as easy as possible?"
Different customers do different things. Some choose to forgo the Product version system in LimeLM altogether, and just use custom license fields to separate their versions (or forgo versioning altogether and use the custom license fields for subscriptions).