This seemingly simple and common situation is actually rather complex and often overlooked. You are correct that if you increase the downstream demand by opening more hydrants the pressure will decrease. But of course the pressure at any point in the system can never go below -14.7 PSIG (0 PSIA) and in practice will not go below the vapor pressure of the water (0.36 PSIA at 70 deg F).
The reason your model shows -20 PSI is that it will assume that it is always liquid water flowing, and will calculate the pressure drop accordingly. What actually happens is that when the pressure gets below the vapor pressure of the water, the liquid turns to vapor (steam) and the flowrate drops off dramatically and the model is no longer a true representation of what is happening in the pipeline.
When you get to this stage and the water is boiling in the pipeline you can get very bad vibration and hammering problems.
My own background is in process plants rather than water reticulation, but in process plants we get a similar situation where cooling water is pumped up to 120 feet or so at the top of the plant, and then has to run down back to the cooling tower. The usual solution to this problem is to install a vent at the high point. This results in air being drawn in if the pressure goes too low. Having air in the system introduces a new problem because you now have two phase flow and the pipes have to be much bigger. And you lose any pressure you had in the pipe. This a whole new ball game which I will ignore for now.
I have seen vents on water pipelines, but I'm not sure if they work on the same basis as what we use in process plants.
Sorry I cannot give you a definitive solution to your problem, but I can confirm that you are correct to be concerned about what might happen. I suggest you call in someone with the right experience in addition to any good advice you might get here.
regards
Katmar