I'll try to answer both questions...
Difficult to answer about best time - depends on where you are going and what you want in a dive.. You can have great viz in one dive site off, for example, Muscat and then crap 10 miles away...what can I say - it's the sea...
Viz is usually better in the summer but then you would have temperatures between 45 and 57-58C which is a tad hot...
Marine life is good all year, water temp ranges from 24 right now (cold as hell-o) and 33-35 in the summer..
Whales..well there is always a chance but probably not, in 6 years here I have seen whales 3 times on the surface and once on a safety stop (8m Brydes whale)...
Off Muscat you'll most likely see dolphins (spinners) from the dive boat but you can't get in the water with them. I only know of a few people who have seen dolphins while diving...
Sightseeing without renting a car means you are stuck in Muscat...go to the traditional suq (market) in Muttrah one evening, spend a morning at Sultan Qaboos Mosque (not open for visits on Thurs-Fri), remember to cover legs, arms (3/4 length is not enough) and hair...Fish market in A'Seeb is different - nice but different... Go for a walk in the mountains in Ruwi..There are a couple of museums worth visiting... Qurm Park... afternoon tea in the lobby of Al Bustan Palace (don't know if they have re-opened though)...
Other people will probably offer you different advice...but with only two days for sightseeing...
Check Lonely Planet's website...
Cheers
Ian