I bet the leak is from the hose back feeding. One way to check would be to hang the hose low to drain it from the head and see if that stops it sooner
Park somewhere with hard water, like anywhere west of the Rockies for a month, and that drip will calcify and dry up
There’s a vacuum breaker in those faucets to prevent the water in the hose from being able to siphon back into your plumbing and contaminate the water. It sounds like you’re seeing the water that’s seeping out from the hose through the vacuum breaker.
It eventually slowed down and stopped but it doesn’t seem right
Hard to tell, is that black part a fitting or the end of a hose?