>> " If I change it to verified trials, can I do so seamlessly without causing problems for my existing/licensed customers?"
Yes. Currently (in TA 3.x) verified trials require product keys, and a custom license field that "labels" the key as a trial key.
In TA 4.x we've added product-key-less verified trials. This is coming very soon.
>> "product keys, and the management of the product keys, work exactly the same with unverified and verified trial versions?"
No, not in TA 3.x.
Yes, in TA 4.x (the same functions you use for unverified trials you can also use for verified trials).
>> " If a customer decides to purchase the product after using a verified trial version, do I then issue a second product key to license the software, or does the key from a trial version change to a key for a licensed version?"
For TA 3.x, with product-key-verified-trials, you can do either. Either modify the pkey to remove the "trial" custom license field. Or issue a new key. It's up to you.
For TA 4.x, verified trials don't require product keys. So, when a customer purchases your app it would be the first time you issue a product key for them.