In order to buy a room a player will need:
-to be level 50
-to have reached the rank of Second Lieutenant in their respective Grand Company
-have 300.000 gil lying around that they're willing to spend on buying the room

Should you relocate your house, those rooms will be relocated aswell - even in their state they're in, like with all furniture in the right place etc.
Obviously the rooms will be lost though, should your FC lose the house or the person leaves the FC.

To answer the second question:
There is no Heavensward housing area, but that aside: as long as you place an Aertyhe (available from your GC and an item you really, really want) in the garden, so that they can teleport to the house, they'll be able to access any housing zone even if they dont have the general area unlocked yet. A level 1 character starting in Ul'dah could teleport to Mist/Lavender Beds/Shirogane as soon as teleport as action is unlocked. I'm only 95% certain, but I think even people who havent bought Stormblood are able to teleport to Shirogane. (Note however that they're not able to leave the housing area for Kugane and cant take the ferry back to Limsa - they'll have to teleport out again)
As long as your people can teleport to the house, you can build it anywhere you want, they'll have access to it!