Key thing is to make sure you give them a flight into Thailand, a flight out of Thailand within 60 days of your arrival, then another flight into Thailand. This is your justification for needing multiple entries. Without this it's possible they issue a single entry visa and keep the difference in price.
There are 2 dates. The first date is the visa validity date. The visa will be valid for 90 days from the day it is issued by the embassy. The expiration date will be on the visa itself.
You must enter Thailand by this date to use your visa.
Once you enter visa, you'll get the number of days granted by your visa, which for non-O would be 90 days. This 90 days will start the day you enter Thailand.
I would tell the checkin agent to mind their own business. They're not immigration and they clearly have no idea what they're talking about.
You're at a high risk of being pulled out of line and being interrogated based on your time in Thailand. You didn't receive any tourist visa, you're received visa exempt, and a lot of people are getting interrogated that spend too much time in Thailand using visa exempt.
You would be best advised to have proof of your ticket out of Thailand as well as proof of 20,000 baht or equivalent in another major currency (IN CASH ONLY) when returning. This last item is the only thing I can conceive the agent was referring to. You might be asked to show proof of money, not pay money to immigration. The only people that have to pay 20,000 baht to immigration are people leaving Thailand with a 40+ day overstay of their visa.
You need to stop listening to this agent whose only goal is to get your money. You don't have to wait for anything to get your LTR. Talk to the BOI, not this agent.