Are any others recommended? Maybe different ops for different parts of the island?
That's probably the case. The ops I've been on tend to stay in/around the bay that the boat departs from.
Do different ops favor certain sites, and are some better than others depending on location?
I think this is the case, ops do have their favored sites, depending on distance, their clientele, etc. On the ops I've been with, morning dives are for certified divers only and are deeper. You'll typically get a wreck first (70-100ft) followed by a shallower reef or drift dive (30-60ft). Afternoon dives include discovery/class checkout dives, and are typically two shallow reef dives.
One Waikiki operator I've dived with does the Corsair and YO a lot. Another prefers Baby Barge and Fantasy Reef. Whenever I've called ahead, I usually get non-committal answers about where they'll likely go, which is understandable.
I noticed while researching online that many operators use the same boats. Does that cause problems, or should it be of no concern?
Not a problem per se, but when an operator simply charters spots on a dive boat, they (and in turn, you) may not have the ability to decide what site to dive. If they have their own boat, the customers have more of a say, conditions depending of course. Also, you may or may not have to put up with more crowded boats on such charters, since the operator again has less control over the total booking.
The last time I was on Oahu, I dove with Island Divers out of Hawaii Kai. They have a big boat, there weren't a lot of other divers on it that day, the sites were good and the DMs were easy-going. I had a good time. And it was the first time in Hawaii that I didn't witness anyone go OOG at depth and suck on the DM's octo during the safety stop. Two-tank morning dives were $85 with pickup/dropoff (I've been quoted as high as $150 there, pre-recession, and another op I called wanted $125, a $30 hike from last year).