Small drip from the T&P valve on my water heater - can this wait until Monday?

Your buddy is right about the missing pipe. Code in most places wants a discharge tube running from that valve down to within a few inches of the floor, so hot water doesn't splash off the tank and scald someone standing nearby. It's not a hard DIY job either. The valve has a threaded outlet on the bottom, you just screw in a length of copper or CPVC pipe sized to the valve outlet and run it straight down. Worst case you need to cut the pipe to length and dry-fit it. Any hardware store has the parts and it's maybe a half hour with a pipe wrench. I did mine myself after the plumber quoted me silly money for the same job.

Ten years old is getting up there but a drip alone doesn't mean the tank's dying. Mine did the same thing around that age and it turned out the valve itself was just shot, not the heater. Swap the valve for like twenty bucks and it stopped. But here's the thing, sometimes it drips cause the pressure's actually running high, and that points to a bad expansion tank or the city regulator going wonky. Worth checking if you've got one of those. If the drip stays steady and the bucket's not filling fast, Monday's fine. If it turns into a trickle, don't wait.

Second reply nails it about the valve being shot at that age. Mine did the same around year nine, dripped only after we ran the dishwasher and showers back to back. Swapped the valve and it still dripped a little, turned out the expansion tank was waterlogged. Tap it near the bottom and top, if both spots sound dull and heavy instead of hollow at the top, the tank's done. Once I replaced that the drip quit for good. Waiting until Monday is fine, just check the bucket every hour or so to make sure the pace isn't picking up.