Heat cycles are not a concern with a street tire. No warmers are needed. Street tires will actually be warm enough sooner on the track than race tires. You still want to be careful for the first lap or so.
Pretty much all tires have the directional arrow. I think that is more to show which way the tire is supposed to be mounted to take advantage of the siping (grooves) for wet weather use. Most manufacturers do not officially condone flipping. Yet, there are lots of track riders that have flipped just about every tire out there without incident. If you want to flip a tire, you need to make sure that you do it before one side is completely worn. Another thing to consider is whether the curve from edge to edge is still even enough. It is no fun to ride on a tire when the profile is shot.
Replacing it is a good idea. It really doesn't have that much life left, and it is nice not to interrupt your day with having to mess with your tire. Also, you need to develop the ability to adjust your riding style as the tire wears. You can't drive out of turns the same as you get down to the last 25% of wear.
If you care to get more life out of a rear on the track, be gentler on the drive out of turns. No power drifting out of turns, that beats up the tire life. But that can take a bit of the fun out of it!
Getting a new tire is a great idea. The other can be sold or given to someone who rides the highways or street, since it has so much good tread in the center. Even for the street, it feels best to have a nice even curve profile from edge to edge, though. For someone who doesn't hit a lot of curves, it would still be fine for the street.
A fresh tire will keep your mind on having fun, and not on nursing the tire, so I say, go for it!