1. When backpacking abroad in a place where I don’t have at least medical insurance, I’ve always used some generic travel insurance with cover for hiking activities. I’ve used This one
in the past, they have some adventure travel product that used to cover hiking.
2. Yes. In France, just write “Poste Restante” in a new line below your name (it’s like the “General Delivery” thing). If in Spain, write “Lista de Correo” instead. I sent my snow tools to the post office in Gavarnie, the HRP goes by the front door and the most tricky passes are east of there. In Spain, they’re supposed to hold things for 15 days but I guess they will for longer if needed if you call them and ask for it. I think it’s the same in France. You can usually send stuff to hotels/hostels if needed if you book with them and ask nicely, I did that at the hostel in Salardu (which is in Joosten’s path and where the PO is several kms. down the road in another village)
Given the terrain, and in my opinion, it’s not a good idea to have crampons and no axe. Poles are no substitute for an axe. Consider some of those passes are really steep, it’s beyond hiking and into (easy grade) mountaineering. If the snow is hard, which can happen, crampons alone may add to the danger.
3. I’m sure it can be done but I’m also sure I wouldn’t be the only one advising you against counting on a GPS as your only navigational tool, particularly in a route like the HRP where sooner or later you’ll need to navigate. I haven’t even tried the GPS thing though.
Good luck with preps and have a great HRP.