1. Can you discover an island using an empty ship? Also do all explorations have to happen at the same time, or can you move+explore with one ship, then do the same with another (all on your turn).
Yes you can discover an island with an empty ship. All explorations do not have to happen at the same time. I believe once you start moving a ship, you do need to complete the movement for that ship. Upon discovering a new hex, the movement for that ship ends. But, you can explore with a new ship.
2. It seems that ship's are also removed when using a settler to build a settlement on the coast, seems a little frustrating to lose the ship too
. If so is this the only time a ship gets removed from the board (not when offloading crew, or just discovering etc)?
The other time they are removed is if you run out of ships. You can actually build more ships if you run out of ships, but you have to remove an existing one from the board. I like to call this "setting fire" to the ship.
3.Setting up the starting neutral settlements in 2P rules seems a bit strange. What are the point of adding these neutral settlements? Could we maybe just use a slightly smaller starting island and not bother with those?
If I had to guess, the point of having the neutral settlements is to force exploration. If neutral settlements were not there, the possibility exists that you could reside solely on the starting island. Not sure about this, though. I think your recommendation for variation has some possibilities.