Building circuit to amplify small sensor signals using an Instrumentation Amplifier and 24-Bit ADC
The output will not swing completely to the rails - with a moderate load it will get to within 0.2 volts of 0 volts and positive supply rail and this sounds to me like a problem because it isn't guaranteed to do any better than this realistically (read page 6 of data sheet "output swing"). 
There isn't a problem here - your signals are DC coupled and they are referenced to 0 volts.



  
  Is it a good idea to work with 2.048V reference voltage in my case or
  would you go with 5V?
  


Go for a higher supply like 5 volts then at least you can guarantee the output can swing up to 2.048 volts.


  
  The chosen ICs seem to be pretty accurate. Do you see huge issues in
  this case going for like 18-19-Bit accuracy/resolution here, with the
  provided system/idea?
  


Yes, I see issues - the input offset voltage is specified in the 100 to 200 uV range and this is an error added to your input signal that you appear to be expecting not to be there.


I don't see a problem here.


If accuracy is a problem I don't see any reason why you shouldn't use a standard rail-to-rail op-amp but one with excellent input offset voltage specifications like the ADA4528. It has an input offset voltage of 2.5 uV and can swing its output to within about 10 mV of positive rail and ground. You'll need two gain setting resistors because it would operate as a conventional non-inverting op-amp amplifier circuit.