Before my customer pays, will the flight API tell me whether that ticket can be cancelled or changed later?

Yes, in most cases you will see the fare conditions before your customer pays, because the flight API sends them back with the fare quote just before the booking is confirmed. That is where you find out whether the ticket is refundable, whether the dates can be changed for a fee, or whether the fare is completely locked once it is issued.

How clear that information is depends on the airline and the supplier behind the search. Some airlines send a tidy summary that your staff can read at a glance, others send long rule text written for airline staff rather than for travel agents, and many low cost airlines simply mark the fare as non refundable and leave it there. The exact penalty is also not always known until you actually ask for the cancellation, so the fare conditions should be treated as a guide to what is allowed rather than a final quote of what it will cost.

On your portal you can choose how much of this your customer sees. Most agencies show a short line next to the fare saying whether it can be changed or refunded, and keep the full conditions stored with the booking so any member of staff can answer the customer later without going back to the supplier.

Run a few live searches on the routes you sell most often and look at what actually comes back for a cheap fare and a flexible one, so you know what your staff can safely tell a customer before taking payment.