Great questions. Depends on the shelf, most of those shelves have wire inserts with either a plywood or poly-iso 'top' (my shelves are usually 30" wide and 8' long - that's how big the tops are), and the water sheds off to the sides or ends. But I still lose the occasional light. This winter I started putting a layer of 6mil plastic over the wire inserts, and then putting the plywood on top of that. So the water sheds off to the ends of the 8' benches. I have only gotten to about half of the benches with the plastic though. It seems to work. I have GFCI breakers on every circuit (in the service panel), just in case.
The mounted plants just get watered with the hose like everything else. I have a Dramm breaker on the end of the hose. I water those every day or every other day in the winter. They are actually a bit behind the plants on the shelves below, so I don't end up watering the potted plants every day.
Humidity is less of a problem than I thought it would be. I don't really measure it, but it isn't oppressive, probably around 80% at at the highest point. I added a reflective vapor barrier to the ceiling (a few years ago), before I did that it would literally rain in the winter (off the roof of the pole barn, onto the plant room ceiling and everything in the unheated half of the barn). It still drips a bit, but is much better. I'm a little concerned about the joists rotting, have to check on that at some point. I've been meaning to install a vent and fan in the 'attic' for 9 years now. But the reflective barrier has minimized the condensation. 3+ inches of insulation in the walls and 6+ in the ceiling helps a lot with the condensation issues.
The floor seems to absorb most of the extra water (and there is a drain). Two corners of the room used to be horse stalls with dirt bottoms (no concrete), I filled those in with gravel and a lot of the extra water goes out that way. The remaining floor is concrete.