Yes you will need the blue adapter for a 16a caravan socket, the mobile charger comes with indoor 10 and 15amp plug ends.
No need for cables at DC chargers
Public tethered type 2 nothing is needed (except the right account/fob etc to activate if required) - this also covers tesla destination chargers
The type2/type2 cable really depends. They are $300 but handy to have esp if you camp a lot or have a lot of public AC charging.