The Souq and Corniche is the best option as mentioned. Very easy to get a taxi to the Souq, then walk across to MIA. You could then walk along the Corniche up towards the Sheraton then get a taxi back from there. I am not sure how late the Dhows go but they will take you across the water from one end of the corniche to the other or just around for 15 minutes and its pretty cheap and a fun way to see all the lights of the city.

Alternatively you could go to one of the hotels for a drink.

With regards to the rainy season, it was around this time last year when there was one big downpour which flooded many of the roads and even the airport roof leaked. I think other than that, it rained on Christmas day and I don't actually think there has been any rain since?!