As far as I know, Al Boom does 3 dives every day in fujeirah from the Al Aqah hotel by speedboat to divesites close to the hotel. They also do speedboat and dhow trips from Dibba to Mussandam (also daytrips). Al Boom does speedboat trips in Dubai, mostly from the Jebel Ali site I believe.
Al Marsa does dhow trips in Mussandam.
Pavillion surely does daytrips in Dubai (from Jumeirah beach hotel).
By the way it seems the waters are clearing up now they stopped the work on the artificial islands. Dubai site is all wreck diving (except at some smaller sites like Sheikh's reef, Russian (jumeirah) beach.
You can also go to some of the other operators (Sandy Beach hotel, JAL hotel Fujeirah (Palms Dive center), Seven Seas divers, Divers Down in Fujeirah, ...)
To answer the question on Nautica Marine Mall: they are located in Al Quoz, behind the Audi Garage I believe. They're a bit difficult to find. They also sell nice boats.