The major problem is that there is an annoying persistant hum that comes through the speakers. We have narrowed it down to a wiring problem because when we take the same equipment and use it off site with new wiring it works perfectly. Also, all the power for our sound and lighting equipment is earthed at the same point - from what I've been reading that causes ground loop problems...I may be wrong but that's my understanding. The hum is annoying but not so bad that we can't use the sound system.
To help locate the problem a little bit more info is needed.
Are the amplifiers in the control booth with the mixer or are they backstage?
Also does the hum change as you change the level of lights especialy when the levels are below 40%. If the hum doesn't change then it's not the dimmers dirtying up the power supply.
Another thing you can try is bring in an electrical isolation transformer. Ie the type to stop you getting electrocuted when working outside. Make sure it has enough current rating and plug the sound equipment into it. This will isolate it from the lighting gear. See if this helps.
You say you take the equipment and it works with new wiring but you don't say what new wiring. Is it just the leads from the speakers or do you change leads from the mixer to the mixing desk.
You don't say where you think the faulty wiring is from mixing desk to the amplifiers ( via any processing). I would start by disconnecting the amplifier input leads then turn the amp up and see if the hum is still there. Then step back to the mixer. If you have any processing gear in between take it one piece at a time. Is the hum on only one channel or both? If it is only one swap leads to see if you can isolate wiring that is faulty.
Does the hum only happen when you have a certain channel on the mixer up. If so what is connected to this channel. Swap the input to a different chanell and re-check.
If you find a lead at fault either replace it or if it is fixed wiring check the soldering on each end. Also use a meter to check for open and / or circuits.
While I was doing this I would keep a log of each step. This way you can go back to the powers to be and say you have found where the problem is. If you can specify where it is they are more likely to pay to repair it. If you just go to them and say it's in the wiring but not where they have no idea of how much it is likely to cost.
Also they will listen to you more if they can see you have thought out this problem in a logical manner.
Also if you can isolate the faulty wire you maybe able to convince the Music Teacher to hold off on a new desk until the wiring is fixed. You maybe able to say tactfully to them that if we get this new desk and it doesn't fix the problem we will look like fools.
Also think of this as a good learning oppurtunity. If you continue in theatre this is the sort of problem you will have to solve half an hour before a show opens.