From the picture it looks like the stucco wall has a stain from the pressure release valve dumping? In any case do you know what the setting is for this valve? If it is 125 psi then I would think this is to high to really provide any protection over time for your plumbing. If it has been dumping then you are wasting water to relieve the pressure.
What is your city water supply pressure to the house and is it being reduced by a pressure reducing valve(PRV)? You want to have around 60psi for the house, so the question is what do you have now? You will need a pressure gauge to hook to that outside faucet to see what the pressure is. Here is a nice one:
http://www.homedepot.com/p/3-4-in-Plastic-Water-Pressure-Test-Gauge-DP-IWTG/100175467
You can let it monitor the pressure for a few hours and see what the max pressure you have over time with that red needle. Meaning you are using hot water and the water heater is being used to cause the increase in pressure in your house. Then again all this is based on:
1.) What is the city water pressure to your house?
2.) What is the pressure relief valve set to on the hose bib?
3.) Do you have a pressure reducing valve? If not then it will be the city pressure and the water heater when it heats water will dump the increase pressure back to the city. But does the city have a back valve in the meter? Then water will not be released back to the city and the pressure relief valve at the hose bib comes into play.
Based on your posts it would seem that right now you have city water pressure in the house and the city does have a back flow valve in the meter. The pressure relief valve at the hose bib is to dump pressure that is greater than what it is set to. This setup works, except you have no control over your water pressure(city pressure) and it wastes water when it relieves the pressure. The best setup is to have a pressure reducing valve and thermal expansion tank and then you can control your house pressure and set it to what you want.
http://www.watts.com/pages/learnabout/thermalExpansion.asp
But, if the city is providing 60 psi and the relief valve at the hose bib is set at 70-80 psi then you can leave as is and the only down side is the wasting of water to relieve that pressure above 60 psi.