Question about GFI in a house
Your problem isn't at all what you think it is

A GFI receptacle has "line" and "load" terminals -- power comes in on the "line" hot and neutral and is fed through the GFI's internal 'switch' (it's really closer to a circuit breaker, just set up to trip on current differential between hot and neutral instead of overcurrent) both to the outlets on the front and to the "load" terminals, which allows you to daisychain lights, outlets, etal off of the "load" terminals of a single GFI and have them all protected that way.

Furthermore, your friend is wrong when he says that two GFI's in series "cancel each other out" -- the configuration of two GFI's in series is legitimate and will not cause a hazard or damage them; it's simply indeterminate which one trips on a ground fault downstream of the second GFI.  Think of it as a game of quick-draw, just with tripping GFIs -- whichever one "wins" trips first and "shoots the circuit dead" so to speak.  (A similar problem arises with regular breakers and short circuits, as well.)

But there is a problem here

However, the configuration you describe is simply impossible without violating other parts of the electrical code -- the kitchen/dining room outlets must be on dedicated small-appliance branch circuits, and the bathroom and laundry room are required to be on their own circuits as well, yet the kitchen counter outlets, bathroom outlets, and laundry room outlets (new for 2014 it appears) all must be GFCI protected.  Point your property manager and electrician at sections 210.8 and 210.11(C) of the 2014 NEC...