>> "1. Am I correct in thinking that if we had a page on our website where a user could request a trial extension be emailed to them, and if we only wanted to give them one extension, that we would need to check via limelm.trialExtension.search if they already had one for that email address before issuing it?"
Yes.
>> "2. Since they could just use a different email address, is there any check if it's been extended, or would I need to use a custom license field with the start date of the verified trial and check that manually before applying the extension?"
Trials do not have custom license fields. Only product keys and activations do.
Frankly, you shouldn't automate doling out of trial extensions. It can be abused without proper "user verification". In practice the number of customers requesting legitimate trial extensions will be a tiny fraction of the customers trialing your app. Small enough to process manually.