The drip mechanism is pretty simple on most of these lamps. To check it just put water in the top chamber while the bottom of the lamp is removed. With the control lever or nob set in the on position it should simply drip water like in the video and photos. Not too complex. If the control lever/nob has a lot of play in it you might end up with problems. The water may drip too fast, potentially dangerous, or might let gas leak back up into the water chamber, non-functional.
You can see the valve functioning in the video. I just made sure it would both drip and close.
It's probably best to use some sort of filtered water in the lamp but any water will do in a pinch. Cave water works just fine as does rain water. Note that particulates will possibly permanently clog your water valve and water in caves is pretty much universally high in mineral content.
I have yet to deal with a broken or loose valve so I can't advise on a fix other than salvaging parts from another lamp. The people I know who use these lamps that have encountered a loose valve suggest turning those lamps into art. Probably more creative uses for a dead carbide lamp but you get the point.
Acetylene can be dangerous. Don't risk injury. Too many old lamps laying around to mess with a faulty one that can't be fixed.
All of this comes with the caveat that if you are good at brazing and pipe sweating you can repair most any of these lamps. Even things like cracked bases and loose gas tubes can be repaired if you know how. Unfortunately if you don't and try without proper instruction you are likely to set your head on fire. ...literally.