How much does a Zobel network and its components affect the sound of speakers?

as "AK" mentioned sounding edgy at higher volume could mean a high amount of off axis output but there are many other potential causes.
I review of your speakers mentions the need for precise room positioning and toe-in.  This also implies that the listening position needs to be somewhat nearfield and also dedicated as lots of true monitor speakers require.  
Does it sound edgy while walking about or in different listening positions other than dead center?  They are designed to sound best listening dead center with considerable distance (4 feet or so) from side walls and optimized toe in.  
Getting the speaker and listening positions at their best is important for these speakers.  

I am experimenting with placement, however, this doesn't address my basic question which is, How much and in what specific way does the Zobel network and its component variables affect the sound of the speakers?

To reiterate, the speakers sound better with the Zobel networks engaged. The networks consist of a Dueland capacitor and a Caddock resistor. What if I changed the capacitor, say, to a Jupiter copper foil and/or the resistor to a Vishay Z foil? As long as the values are equal, would this change the tonal characteristic of the drivers or internal crossover, and if so, wouid it be a negligible change or a palpable one?

You definitely need the zoebel and might see an improvement in sound with new parts if the current cap and resistor are out of spec or defective.  
the zoebel is not a critical part of the crossover network that would lend itself to sound quality upgrades with improved quality parts.  
If you have DIY capabilities those parts should be inexpensive all things considered if you are determined to give it a try.  

Caddock MPs are 10W resistors.  Vishay/Texas Components Z-foil are .25W.  If you want to experiment, use 10W alternatives such as Path, Duelund, Mills, or ebayed Russian SSG silver mica.