Car: 99 Civic Si

Problem: I bought the car from a dealer in December, and have been having some little problems here and there.
1. The battery (looks stock) has the little green eye/black eye indicator to show the life of the battery-green being good, and black being insufficient. The eye is BLACK, but I got the alternator and the battery tested a couple days after I got the car and they both tested good...?
2. While driving at night, when I use my turn signal to change lanes - my headlights and taillights flicker really quick and then come back to normal brightness. The lights also dim when I use the power windows...like when I pull the button up to roll the windows up - the lights will dim until I release the button. I was thinking it may be a bad ground cable? Fuse? Battery? Im not sure, and would like to get this fixed soon, so that it wont cause any further issues.
3. When I have the heat on full blast and on defrost-the AC automatically kicks on...is this normal? Like Ill turn on the heat and the AC button on the dash will light up, and when I look under the hood the AC will kick on for about 4-5 seconds in 30 second intervals. I have never owned a car with AC, so this is new to me.

I plan on getting a new battery tomorrow, so we'll see if that helps any of these issues. Thanks in advance.

-Josh