Leena,
As you can see there are a variety of opinions. Personally I like the Benwood slightly better than the City of Washington, but I would do either in a heartbeat. I haven't been fortunate enough to see the sharks at the C of W, but I have seen a lot of fish at the Benwood. There were some really huge Parrotfish at the Benwood, plus Baracuda and schools of grunts and yellowtail snapper when we were there. I am sure that timing and luck have a lot to do with what you see. Also the turtle "Bubba" hangs out at the Benwood every night.
I also liked the Benwood better because it was slightly deeper. My dives on the Benwood are always 38-39 feet, while the C of W was only 25 feet. Personally I like being a little deeper, but that is just me.
Usually the dive boats try to go to where the conditions will be best, and to match the skills and desires of the clients (that's us!). However, they don't like going to the same old places everyday, and don't like going to the same place in the afternoon that they went in the morning.
As for "requesting", it depends upon the operation. I would definitely suggest asking for what you want. The worst that they can say is "no" and maybe they have good reasons. In my limited experience, I have found that the dive boats want happy clients, so they will try to accomodate you. Sometimes the destination is set up in advance, but not always.
In 2002 we asked the Captain as soon as we got on to go to the Benwood. When he told the DM that he was going to the Benwood, she made an ugly face and asked why. He said that he "had requests for the Benwood" and that was that. That particular day the boat had divers and snorkelers so the DM wanted a shallower reef. If we hadn't asked, we would not have gotten it.
Asking early is recommended. Once they have made up their minds, or made promises to other clients, it is harder to change their minds.
Of course you can always try to ask the dive shop in advance, but usually the final decision is made at or near the departure time because conditions can change.
We also dove Mike's Wreck at Key Largo. It obviously didn't leave much of an impression because I don't remember anything about it. My online log says it was 28 feet. My actual log book with any more detail is at home.
Of course the Speigel Grove is an awesome place to dive, but it is a completely different beast from the others discussed above.
Just my opinions, and only because you asked.
Wristshot