DC bike questions from a new DC resident

Hi, just moved here last month and recently started biking with a road bike. Had a few questions about biking in DC. I am also new to biking on the road in a city.

1. Do we have to completely follow traffic lights? e.g. if i am first in line at a red light but i can safely deduce that i can cross while the light is red, can i cross?

2. what if i was on the sidewalk can i cross the street when i'm facing a red light but it looks safe to cross?

3. is it proper etiquette to stay in the right most lane on the road unless you need to turn left?

4. what if there is a right turn only lane and you want to go straight at a red light but you don't feel comfortable riding in the middle lane? (happened to me the other day on a capital bikeshare bike... was in the right turn lane wanting to go straight and a lady honked and yelled at me)

5. how safe is it to haul your bike on a bike rack strapped to a car but is not locked down? i live by eastern market and the other day i saw a guy steal a bike from another guy in broad daylight (7pm) by the eastern market metro stop about 100 ft away from like 5 cops sitting on their motorcycles having donuts outside the dunkin donuts. the bike was retrieved after someone in the street knocked the thief off the bike (cops tried to unsuccessfully run after the guy) this does not make me feel comfortable with hauling my bike around town on my car since my bike rack doesn't lock to my car nor does it have locks for the bike.