Yes, probably. Driving with the lights off puts you at greater risk of a collision, so a greater risk usually translates to higher insurance rates.
If they find out about it then it might.
No. It is a nonmoving violation.
If it's your first ticket, you will most likely be able to go to traffic school and then it wont show up on your record.
None, if you take traffic school.
Generally, your insurance rates will only go up after receiving a moving violation. Just consuming alcohol in a public place shouldn't raise premiums.