Agree with David, the Japan Guide website is very helpful.
I-The airport is in Osaka. So you do take the train into Kyoto. It is easy to do. Go upstairs at the airport to the JR ticket station. You can buy a card that includes the Kyoto subway. They take only cash. You can get more details on the Internet.
II-Avoid the winter and summer months. Miserable weather. April in Kyoto might be busy because of the geisha dances. May is OK but avoid Golden Week. September is the tail end of monsoon season. Late October and November get busy again with autumn colors and students on school trips. That leaves the first part of October as the best time to go, IMHO. No major holidays or events and the weather should be nice.
III-There's no real city center or downtown. But some people like to stay around Kyoto Station because it is easy to catch the subway to other places. Plus, there are loads of places to eat and shop inside the Station. Or in Gion for the atmosphere. Taxis are plentiful and inexpensive, so hotel location is not so important. Carry your hotel business card. They're usually English on one side and Japanese on the other. Just show the card to the taxi driver.
IV-Hard to say. You can eat cheaply or expensive. Unlike the States, their chain restaurants have tasty food.
V-David answered that one.
JapanesePod101.com has free language lessons. Plus, there is information on culture and etiquette. However, you can get by without Japanese at most major tourist places. Restaurants have English menu. Or there're the plastic models in the window, so you can just point. People are very helpful and friendly. The city is clean and safe. You will love it.
If you have the time and like a challenge, take a Japanese course at your local university. At my university, senior citizens get a discount. Only $5/credit. So a year of Japanese was $40!