Your airport terminology is confusing! What are the two Ryukyu Airports? I have never heard the term.

The main airport on Okinawa Island is Naha Airport (also known as Okinawa Airport). And Miyako's airport is commonly known as Miyakojima Airport rather than Hirara Airport.

It is difficult to answer your question without knowing what are the two "Ryukyu Airports". Ryukyu refers to the entire island chain that constitutes Okinawa Prefecture.

1st time to Okinawa, I think there are more to see/do on Ishigaki and Iriomote than on the main island

That is not true. The main island has more to do than any other island. But for the best nature, I recommend Iriomote. For the nicest beaches, I recommend Miyako.

What are the names of the domestic airlines that fly within Okinawa, say from Ryukyu airport or Hirara to either Iriomote or Ishigaki.

ANA and JAL (including Ryukyu Air Commuter and Japan Transocean Air).

My wife doesn't like boats as she gets seasick so we would need to fly between Ishigaki and Iriomote.

If you want to go to Ishigaki, you must take a paid flight. You can connect at either Naha or Miyako.

I have been to both Okinawa Main Island and Ishigaki and I think that Ishigaki is nicer than Okinawa Main Island. Ishigaki has wide, beautiful beaches, some historic sites and a lime stone cavern. You can go by tourist taxi anywhere on the island. Miyako is nice, too.

2. What are the names of the domestic airlines that fly within Okinawa, say from Ryukyu airport or Hirara to either Iriomote or Ishigaki.

From Naha, you can take either ANA or JAL to Ishigaki. From Miyako you can go to Ishigaki with JAL only.

3. My wife doesn't like boats as she gets seasick so we would need to fly between Ishigaki and Iriomote.

If you fly by ANA from Tokyo to Ishigaki, you connect at Naha (OKA).

I don't remember how bumpy the travel to Iriomote was, but I don't think it was very bumpy. My girlfriend at that time also didn't like to go somewhere by ship, but she didn't get sick.

The boats to Iriomote are high speed boats. They hurry through the sea at high speeds, resulting in quite a bumpy ride. The degree of bumpiness depends on the height of the waves. If the sea is calm, the ride is also quite calm. But when the waves are high, it can be an extremely bumpy ride. I once went shortly after a typhoon when the sea was very rough, and it was the most scary boat ride I have ever been on. But on another occasion, the sea was calm, and the ride was little problem. Still not the smoothest of boat journeys, though...