The tube on the battery is a vent tube. It should not be connected to another tube, or anything else. The battery should not have any acid overflow. Was it possibly over filled with water? The cells will have a fill line.
The tube in the second picture looks like the carb vent tube.
2006 Ninja500R Purchased new July 2006; 0 miles. Miles as of January 2020; 108,827 . It's a GO bike, not a SHOW bike.
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