You might as well get some local cash from the airport ATM after you clear customs. I'm not sure if the taxis take credit cards. My sense for cab rates is that they are similar to what you see in NYC or SF. Perhaps others here will give you a more informed estimate for your cost from DXB to your hotel. My wild guess: the equivalent of US$55. The driver might not speak much English, but it will be enough for your purposes.

Yes, this will be safe. I would imagine that Dubai has one of the lowest crime rates in the world.

As for attire, it's all over the map in Dubai. In the same restaurant you might see a conservative Muslim woman wearing a burqa and a Russian tourist dressed almost like a hooker. I recommend no shorts except to swim/exercise. In the evenings, dress like you're going to a cocktail party in SF. It's probably not going to be crazy hot when you're there.

Passport: no need to carry it with you when you're out and about. I kept mine locked in my hotel safe.