If you increase the address space for an OS, does the Phys Mem used by the program increase?
No.  Due to the way virtual memory works, you can have more virtual pages than physical pages (frames).  Some virtual pages might not be mapped to any physical page.

It doesn't really make sense to talk about "if you increase the address space size, the number of pages increases".  It might be slightly more useful to think of it as "if you increase the number of pages, the address space size increases".

But really, in any given architecture, the address space size is often fixed and cannot be modified: the size of a virtual address is fixed and cannot be changed, and that determines the number of pages and thus the maximum size of the available virtual address space.