You are correct: Utility companies do not have to make their installations compliant with the NEC, nor do federal government facilities. In most cases, utility standards are much higher than those mandated by the NEC and government facilities are still NEC compliant even though they technically do not have to be.
As for the green hand hole box, those are usually meant for housing underground sprinkler controls. There's not much effort made to make them even remotely watertight since there's no need to keep water pipes dry. I would be willing to bet that this isn't part of the original installation. There was probably an above-ground J-box that kept getting hit by the grounds keeping people. A handyman probably took a trip to Home Depot and came up with your current "solution".
A true Quazite box will be somewhat more tightly-designed to keep as much water out as possible. Of course, with an open bottom, there's not much that can be done about rising water from the bottom. You can dig down as far as possible and fill the base with gravel to aid in quick disbursement of collected water though.
Of course, the best thing to do is to situate the splices as high up as possible so that the likelihood of their sitting in standing water will be minimized. If the wires are solid, they will usually stay upright on their own. When stranded wire is used, I'll sometimes just bang a short piece of 1/2" PVC pipe in the ground and secure the wires to it with a cable tie or two. If this still concerns you, you could always get a PVC junction box that fits within the confines of the hand hole. You can then extend the existing conduits into the bottom of the box. If they won't all work into that plan, maybe you can cut some of them back a bit and extend them into the proper position.
It's true that THWN wire is rated for wet locations, but that doesn't necessarily mean that people should go crazy with assumptions. My guess is that it wasn't necessarily the wire insulation itself that caused the issues you are having. I'd imagine that the splices being submersed in water cause enough corrosion to result in excessive heating during dry conditions. Over time, this heating caused the insulation to deteriorate.
If you decide to keep the box as-is, just try to keep the splices as high up as possible and also try to situate the wire nuts so that they are vertical (wire openings facing down). At least no water will get in due to the cover leaking and any ground water rising will have to come WAY up. Even then, it might not stay very long if you put a decent amount of gravel in the bottom.
