Re-routing discharge pipe from hot water heater's T&P relief valve
Draining up does not work

The T&amp;P valve drain pipe is no different than any other drain, in that it relies on gravity to function at the end of the day.  As a result, your builder's completely bizarre installation needs to be removed and redone, as you show in your pictures, as it violates several of the operative points in IRC P2804.6.1, including points 9:


  
  Be installed to flow by gravity.
  


and 12:


  
  Not have valves or tee fittings
  


PEX is listed in IRC Table P2906.5 as an allowable material for water service, though, so Point 13 permits its use for T&amp;P relief valve piping.

Your plan might work; you may need a pump, though

Your plan may or may not be acceptable, it depends on how the AHJ interprets IRC P2804.6.1 point 6 in your (admittedly odd) case, though:


  
  Discharge in a manner that does not cause personal injury or structural damage
  


If draining, via the obligatory airgap of between twice the pipe diameter and 6", to a floor that is not further drained is not acceptable, what you can do is use a basin and float-operated pump setup so that any drainage can be pumped to a place that doesn't risk damaging the house.  (They make all-in-one kits of this nature for draining things like washers.)