How can my fiancé extend his stay in Thailand after entering on a visa exemption?

He might need an airline ticket to leave withing the first 30 days as he's entering on a visa exemption depending on airline

1. My first question is how long after his arrival can we apply for the extension?

-> Depends on the province he will be staying at the time of application. Bangkok, Chiang Mai, and some others allow you to apply for a 30-day tourism extension when you have 45 days or less (means anytime once he arrives there) with the entry stamp. Most offices accept when the stamp remains less than 30 days. But a few offices including Jometien (Pattaya) and Samui let you apply for tourism extension only within the last week of your stamp. Check the local immigration office for their policy.

2. Can we extend the Tourist visa another 30 day?

-> Tourism extension is only ONCE per entry, then you need to leave the kingdom.

3. Will that work to get another entry visa exempt?

-> You can apply for a 30-day tourism extension ONCE per visa-exempt or tourist visa entry.

He needs to leave one more time to stay until later July as a tourist.