Sigh. I did understand what you meant about how and when the buzzing started. Based on your additional information and testing, my assumption is that, after some (variable/unknown, as you now point out) amount of time, the OBi gets "fried" by something other than normal telco voltages on the line jack. Given that your basic voltmeter test results are normal/nominal, then I suggest that you do as Rick said: plug a known-good analog telephone (preferably one without its own power supply/AC adapter) into the phone jack, plug the OBi's line jack into the NID, and let it "cook" at least overnight. What's the rush? Take the time to prove that the device itself isn't just a crappy production run, before adding more complexity to your situation.
If it dies, call Obihai, and demand to get some help from them -- suggest that they have a big problem on their hands, and they, not Newegg, RMA-swap your box and send you another one, and ideally, they test the failed box to see why it failed.
I would also recommend that you call Verizon repair service again. Tell them that you have been frying some equipment when plugging into their circuits, and ask them to send a truck out to your location to test the NID for a possible ground fault. If the NID isn't properly, tightly connected to a reliable, true ground per the National Electrical Code (like a cold water pipe or a ground rod in the earth), then you can get ground loops or static discharge on the phone lines.
Depending on the age of your residential 110 AC wiring and the competency of the person who installed it, you may also have one or more outlets that are not properly grounded, or the neutral (white wire) isn't properly bonded to ground at the electrical service panel. You'd need to hire a licensed electrician to test that, or at least buy an outlet tester at the hardware or big-box store. This isn't likely your problem, but since it's been brought up repeatedly, you might as well get some peace of mind. And, speaking of house wiring, it is not unusual to find that the phone wiring, or even the AC wiring, has been punctured somewhere inside a wall by a nail, drywall screw, wire staple or rodent teeth. My house was built in 1989, and it has a flaky line 2 pair of an 8-wire bundle, somewhere inside a wall, likely caused by a puncture.
As an aside, regarding voltage testing and safety: the better/safer way to test phone wiring (vs. the "sewing needle" method) is to cut the modular plug end off of a phone cord, strip off a couple inches of the grey jacket, and strip each of the 4 wire conductors. Then, use an alligator clip to attach your voltmeter leads to your new "phone wiring test cable". On-hook voltage is, as you found, about 50 VDC. This won't kill you, or even hurt much. Off hook is much lower, since there is now an impedance across the line that lowers the voltage. It is the RINGER voltage that will hurt, since it's ~100V AC, and, as Thomas Edison used to demonstrate by electrocuting small animals, to try to scare people into fearing his competitor's AC power, 100V AC hurts more than DC, but only during the brief ring cycle. You wouldn't die, unless you were sitting in a bathtub, though...it's just one of those OWWW $#&%$! moments.
RE: phone cord wiring: all telephone cords should have the 4 wires crimped to the RJ-11 jack on one end in the opposite direction from the jack on the other end. Throw away any that aren't built like that. Ethernet RJ-45 jacks are wired straight through, unless, of course, it's a "crossover" cable.
So, using the photo of the RJ-11 jack in this thread, one end should be 1-2-3-4 = B-R-G-Y and the other end should be Y-G-R-B. Modern telephone equipment doesn't care about polarity, as did the old Western Electric phones, but you never know if some weird phone device DOES care, and the cords are so cheap that there is no reason to keep any cords that are wired bass-ackwards. If you are a cheapskate, or you want to make custom-length cables, you can buy a crimping tool and a bag of RJ-11 jacks and crimp them accordingly. If you have a HP OfficeJet or similar fax machine, be sure it is connected to the phone line with a 2-conductor, not 4-conductor cord. I don't understand why, but this has always been a requirement by HP, and I assume they are doing something non-standard with the outer conductor pair.