Well, that sounds... exciting.
Yes, the walls, posts, stringers and tiles all have a role in supporting the raised floor. I have always used the rule of thumb that no more than 8 contiguous tiles should be removed from a row at once. So, unless you take structural precautions, your method will probably collapse the floor. I'd discuss options with several floor contractors or engineers before deciding what to do.
As for running cold water overhead, this runs the obvious risk of leaks, but also condensation. The risk of small leaks can be mitigated with gutters, but a pressurized leak will go everywhere.
My advice is to try to keep or replace the raised floor as a pathway for power and chilled water piping, even if you do opt for some type of above floor cooling.
My 2 cents.
Ken