Slow drip at the top of my gas water heater - emergency or can it wait?

The green and white crust is the tell here. That's corrosion from water that's been weeping out of that joint for a lot longer than this morning. By the time you see buildup like that, the fitting or the connection underneath is usually pretty far gone, and a drip rate doesn't tell you much about when it lets go. I've seen slow weepers hold for weeks and I've seen one let go overnight and dump 40 gallons in a hallway. So no, you prolly don't need a middle-of-the-night emergency call, but this isn't a 'couple of days and see' situation either. Get someone out within a day or two at most, and be prepared for them to tell you the tank is done. Rust on top of the tank around the port is not a great sign, because if the tank itself has rusted through at the connection, no repair fixes that.

I'd push back on the 'tank is done' part. Crusty buildup around the top connection is usally the dielectric nipple or the fitting itself weeping, not the steel of the tank rotting through. I had the exact same crusty mess on mine a few years back, tech swapped the nipples and the tank had plenty of life left. Not saying that's guaranteed here, but assuming the worst off one rust spot is jumping the gun. A decent plumber will know the difference in about two minutes once they get eyes on it.

Did this exact job myself last winter on my gas unit. Cold inlet fitting was weeping and I figured why pay a service call to swap somthing I can wrench. Two things I learned: kill the gas to the heater first, and shut the house water off and open a hot tap before you break that connection loose or you'll get surprised. Mine was just the fitting, threads were fine, fresh nipple and it's been dry since. Only snag was clearance. That flue sits close and I couldn't get a straight pull on anything, ended up using a short pipe wrench and going slow. If yours is tight too, measure what you can reach before you start unscrewing things.

To your last question, I wouldn't shut the water off at that drip rate. A bucket and towel is how we handled these for years, and cranking an old shutoff valve that's been sitting open for a decade sometimes starts its own leak at the packing. Leave it open. One thing though, don't pick at that crusty buildup before whoever comes can see it. It's ugly but it's half what's holding the weep back, and scraping it loose turns a drip into a stream fast. Take a photo of the rating plate on the side of the tank while you're down there too. The year stamped on it settles the repair vs replace arguement before the tech even pulls up.