On the traveling circus RAA on our way to Cody is was so dad blained hot I can't believe the tires didn't melt. Coolant tank was empty then and no overheating. I put coolant in it a couple times early on. It seems to just piss it back out. So I left it empty. Been that way for about 65,000 miles.
As for the battery, I still have the original 2005 battery out of my bike. I keep it under the bench for test power. I charge it a couple times a year. My guess is it would probably still start the bike unless it was really cold out. They must have missed putting the "three year failure tablets" in that one.
The line that connects on the overflow tank by the cap is the holy $hit overflow line. It dumps to the ground behind the engine near the battery vent tube.