If you want your plant to grow above the balcony edge, you need one that grows tall but is pretty resistant to wind and sun. How about bamboo in a planter. They grow tall and do a great job (all year round) of hiding what is next door. They are fairly low maintenance and self-sufficient once established. Cons: They do cause a lot of leaf drop - some species look pretty awful in the winter... but you can find some that are beautiful. Go for a clumping bamboo.

I'm not 100% sure but those look like morning glory to me. http://m.wikihow.com/Grow-Morning-Glories

Eventually when you come to resell, a pool is an added plus. But for your own purposes, emptying the pool and adding a deck over it with a hinged door and steps placed below would be great for storage (as per option b but bearing in mind you could revert). All the best for your lovely new home.

Same happened to me and I still regret not saving it on time. I later realized I could have just cut the healthy parts and propagated it. Can find a lot of video and material about propagating from leaves and stem cuttings.