Electrical – What safety equipment is best for live multimeter testing on 240v

Safety glasses and ear plugs (and work gloves too).

Get a good pair of safety glasses so they are comfortable, won't fog up, and do not scratch easily and then take good care of them (just like they are regular glasses). Wearing safety glasses when you are working below anything is huge to not only protect your eyes but also just make it so much easier to do the work. If you are working outside then get the sunglass ones (they can double as just regular sunglasses too - I leave them in my car).

Buy a package of the foam ear plugs... and use them all the time. This is the other item that is huge to protect your hearing but then also make it much much more comfortable to do whatever loud do-it-yourself job you are working on. You will notice a big difference between your stamina when wearing the ear plugs. For instance, if you are pounding in nails, with the ear plugs in you won't notice the sound, but without them, the constant high pitched noise takes a toll on your ears (and wears you out for the job faster). (They also are great for just blocking out noise to let you concentrate.)

The other basic safety equipment is work gloves.

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.