Who is responsible for my condo balcony?

You have to check your building's regulations. Generally, they want a uniform look to the outside of the building. I've seen a number of buildings where the balconies were repaired by the condo/coop as part of the mandated façade work.

Should be condo as it seems to be a part of structure of the building. Unlikely they will let you change the railing.

Typically a balcony is considered a Limited Common Element - which means you have exclusive enjoyment of the space, but the building is responsible for maintenance. Check you bylaws for confirmation of this. So not only will the building be likely to be responsible for the tiles- but also the railing. Keep in mind there are specific regulations for balcony railings in terms of minimum height as well as for structural stability. That said, it's likely the building will not allow you to switch out your railing on your own. It sounds like the south facing balconies are having drainage issues which might be causing the issue with the tiles. Your board should have a qualified engineer review your balconies (though this might already be in process in order to comply with the FISP - assuming your building is above 6 stories).

I don't understand the reference to the Sopranos-esque railing - is that some sort of comment about the possibility that an Italian-American might throw you off the balcony?
And what is this about the ghost?