I'm going to make this point loud and clear.
DO NOT TEST THE UPS BY UNPLUGGING IT
You break the ground, which means if any of the hardware has a short, and there is no other ground it will go though you to get to ground if you end up being the shortest path. If everything is working fine this will never happen, but hey if this never happened there'd be no need for a ground.
Best way, is to have the outlet that the UPS is plugged into able to be switched on and off so that the gound and netural will remain intact though the test. The breaker can do this or you can do local high quality switch. If you can't do that, then put a VERY GOOD (ie $30 to $60 range) power bar with an off switch between the wall and the UPS, make sure you label the switch for what it is for. The point in the other post that is mentioned in the comment is to NOT over load the powerbar, doing it this way is better then unplugging it. You can now switch off the line in and simulate a power failure, this will leave the ground and netural intact.
You can test by letting it run down, although crude it will work. Also if the software has a calibration option it will do that for you and run it ever 6 months or so. The run time will degrade over time, so if you are using monitoring software to shut down the servers at say 15%, that 15% will change over time and it can correct for that.
For your voltage issue, if you can run a decided line from the power box to the servers and us it only for the UPS. Things like tube lights, fans, motors, etc will create dirty power. Having the server on its own will help that since it moves it to a circut further away for those items. If its still happening it might be worth getting an isolator put in or it could be that your utility power is just really bad. Put a good meter on to a line and see what its really reading. It needs to be a good meter because I've seen cheap ones be off by 5 volts and thats enough to cause a UPS to go into over voltage so you need an accurate number. If this is a church, there is a chance you have an electrican as a member that could help out.
Here are reference links to grounding and daisy chaining UPS