Electrical – GFCI outlet reading between 80 and 640 volts on multimeter

It is difficult to test without a ground reference. One way to do this is with a very sensitive (very high impedance) voltmeter such as a digital voltmeter. Make absolutely sure you are not grounded (and no pets or children are around). Make sure the digital voltmeter is definitely on the voltage setting. Plug one probe into one hole of the outlet, while the other probe is stretched out some distance. There should be a "phantom voltage" reading of somewhere between a few volts to nearly the full line voltage (for you in North America that would be around 120 volts) when plugged into the hot side. On the cold (neutral) side, there can also be some phantom voltage but it should be much less.

There is a potential harm in having the polarity reversed. Lamps with screw in bulb sockets should always be polarized so the rim of the socket (which is more easily touched) is connected to the cold side. That way to be shocked you'd have to stick your finger all the way inside the socket (you would not do that). With the polarity reversed, the hazard exists at the rim, which may have some exposure even with the bulb all the way in.

The hazard can also exist with older appliances that have the cold side intentionally connected to the appliance metallic frame. That was the only way to make them safer before the grounded outlets became widespread. But it required the outlet polarity be correct.

You can mitigate the hazard by using a ground fault current interrupting (GFCI) receptacle even without a ground wire available. These devices do their intended function even without a ground wire attached. The test button also should work (it cheats by running its 5ma current test between hot on the load side and neutral on the supply side ... not ground). If you find a device where the test function fails to trip it, do not use it. Mark the device as "UNgrounded" for circuits with no ground available. Do not use an alternate ground source that is not part of the supply cable for this purpose. Never use the neutral as a ground.

The most common hazard to people is being grounded when coming into contact with one wire that happens to be hot. The GFCI will trip in this situation even though the current is as low as 2ma. If a metal appliance has internally faulted where a hot wire has nicked or damaged insulation and is contacting the frame, this hazard exists with the frame of the appliance. In grounded situations the ground wire would cause that to short circuit.

Best I can tell, it's time to buy another GFCI (the correct one, this time -- hopefully, the third time's the charm)

There are four problems here:

1. The old GFCI finally died (at power on, likely) and the new GFCI is likely DOA. Easiest fix is to try getting another GFCI and putting it in.
2. Your new GFCI was the wrong part -- you need a weather-resistant receptacle in a damp or wet location these days (NEC 406.9), and yours isn't. Leviton's W7599 is the equivalent of what you have in a Weather Resistant model, BTW (as of this post, at least).
3. Receptacles in wet locations require a while-in-use weatherproof cover or "bubble cover" these days, such as this. This helps keep rain out even under wind-driven conditions, no matter whether stuff's plugged in or not.
4. Even weatherproof boxes need to be pressure equalized (via a drainage hole in the bottom) in order to keep pressure differences from driving water through even the slightest imperfections in seals and to give any water that does get in a path to escape. This is allowed as per 2014 NEC 314.15:

314.15 Damp or Wet Locations. In damp or wet locations, boxes, conduit bodies, and fittings shall be placed or equipped so as to prevent moisture from or accumulating within the box, conduit body, or fitting. Boxes, conduit bodies, and fittings installed in wet locations shall be listed for use in wet locations. Approved drainage openings not larger than 6 mm (¼ in.) shall be permitted to be installed in the field in boxes or conduit bodies listed for use in damp or wet locations.

See also BSI-004: Drainage, Holes, and Moderation for a good explanation as to why even the most snugly sealed enclosures need weep holes. So, if there isn't a small hole in the bottom already, drill a ⅛" hole in the bottom of the box, as per NEMA bulletin 110 and the aforementioned NEC clause.