How can I prevent ground wire buzz in my wiring?
Yeah, recording gear + gypsy electrical is a recipe for more kinds of disaster than just buzzing.   

On the other hand, a carefully set-up system can provide for all your needs with both safety and capacity, even with existing wires.  Mostly it's a matter of getting details right, not expensive gadgets. 

First, a "tree" topology with several branches is a normal way to wire a circuit.  When trimming back the excess electrical branches, no need to tear the wire out, you may regret it later. Just disconnect it inside the box, cap it off with a smallish wire nut, and wrap it with electrical tape so the cap doesn't fall off.   Anyway, unused branches won't cause buzz.  

Second, check out the grounding system.  look for 2 features: 


a ground wire run all the way back to the main panel, to protect you from electrical faults, and if your wiring doesn't have that, that's a problem but we can fix it definitively for about $200. 
a grounding rod, to protect your gear from lightning, static electricity etc.  It is optional if there's no breaker panel. 


Safety Options

Now if your system is not grounded, the first safety feature you really will want is a GFCI protection device, and I want you to use a deadface unit.  Why? It forces you to pause to understand how GFCI's actually work, which means you'll be able to protect the whole place with one of them ($25) instead of several.   

A GFCI won't help with hum but it'll do worlds of good for safety. 

The extreme option

I don't normally go to this option so quickly, but an isolation transformer was suggested, and why not go whole-hog and lick your grounding problem too, while also upgrading circuit capacity.  Giving a modern, state of the art service. 

Instead of a 120-120 isolation transformer, find a 240/480-120/240 supply transformer.  A 5 KVA unit should suffice, and I see them used for $100.    It takes power from the old supply cable, and isolates it, giving 120V or 120/240V at the shed.   Since it's isolated, it doesn't need the ground wire from the house, just the grounding rod. 

If your supply is 120V, you can feed that onto one 120V secondary and draw from the other 120V secondary.  Or, you can change the supply to 240V and put that into the primary jumpered for 240V, and draw 120/240 split-phase off it at twice the power.  (you could even punch it up to 480V with a second transformer, if you have a long distance to go.) 

Double extreme

As ThreePhaseEel notes, the problem is often your own gear.  But I'll discuss your old fluorescent lights and things like that.   You can get a second supply transformer and use it to supply the non-music loads.  This would put 4 windings between your lights and your music.