Can the flight booking API show me how long the stop is between connecting flights and whether my customer has to change airport?

Yes, it can. Every result comes back with each leg of the journey listed separately, with its arrival and departure time, so your booking screen can work out the gap at each stop and show it to your customer in hours and minutes. The airport for every leg is listed too, so when the flight lands at one airport and the next one leaves from another airport in the same city, you can see that before you sell.

What you do with that is your choice. Most agents set a shortest connection time and hide or flag anything below it, because a stop that looks fine on paper can be hard work in a large airport where your customer has to clear immigration, collect a bag and check in again. Airport changes are worth showing clearly as well, since getting across a city means a taxi or a train and time your customer has not planned for.

The part to be careful about is a journey put together from two separate tickets. When the whole trip is on one ticket, the airline is responsible for your customer if the first flight runs late and the connection is missed. When the API has joined two fares into one itinerary, a delay on the first flight is your customer's problem and the second airline does not have to do anything about it. The search result normally tells you which of the two you are looking at, and that is worth checking before you confirm anything.

Decide the shortest stop you are comfortable selling, set that as a rule in your search, then look at a few long journeys with connections to see how the stop time and any airport change appear to your customers.