That many drips and it's still hot. Yeah, that's weird.
Do you have any legitimate reason to think the guy that did it knows what he's doing?
It is weird that it drips but still runs hot. But have you (the OP) confirmed that the drips are coming out the front of the stuffing box (i.e., where the shaft exits the packing nut toward the transmission) or are the drips coming out of the back of the nut? If the former, it's hard to see why the nut would be so hot. As you look at the shaft where it exits the nut does it look moist?
Since the shaft runs at a downhill angle, drips from the front of the gland will run downhill on the underside of the box and then likely drop down at around the aft end of the box. But that's where they will also drop down if, for example, the nut isn't tightened properly. So I could imagine a situation in which too much or too tight of packing was jammed into the gland, which would make the shaft run hot on the front end but would leak water out the back end. If you are just focusing on the drips you might not have noticed their point of origin.
My suggestion is to repack it with the proper sized packing. The packing should be snug but not overly tight. If you have to hammer it in then it's much too tight. You should be able to press it in without a great deal of difficulty. But there should be some friction so it is not flopping around loosely, either. I wish I could quantify it, but I'd just say "snug, not tight," if that tells you anything.
If you use the dripless packing (recommended), then adjust it so it is "drip less" but not bone dry. A thin film of water (at least) is needed for lubrication Buck Algonquin, one of the major stuffing box manufacturers, specifically says that some moisture is needed regardless of the type of packing used, including dripless. An ideal adjustment would be no drips with the shaft not turning and a slight weeping of water (out the front of the gland) when rotating. This should be sufficient to cool it. Again, consult the thread for which I provided links earlier as it contains some good info. Also, MaineSail has a good tutorial on how to pack a stuffing box on his excellent Compass Marine site (
http://www.pbase.com/mainecruising/stuffing_box)