Turntable Hum

Yes, inherently noisy, and this can vary a lot between cartridges and arms even on the same phono stage. Usually anything low level and uniform like you're talking would be within the norm. You can try and reduce if it matters to you but could spend an awful lot of time for very little to gain. Except when it is one channel like you said. In that case you should be able to get the noisy one down to the same level as the other- unless there is something wrong somewhere.

That is what we're trying to figure out. That is why you want to try the phono stage somewhere else. If you get the same noises, it's the Allnic. If not, keep looking. 

You picked up on the main idea, phono stages have a tremendous amount of gain. Things that would never be audible anywhere else can be unlivable with all the gain. Worth it in the end, even if it doesn't seem that way sometimes.

Power cords themselves won't cause this except that they are the path to ground. So lifting the ground on one or both is another test. Use a cheater plug to test this. Then if that doesn't work I would try the Allnic somewhere else, another system, anything at all just to rule it out. 

Solving puzzles over the internet is a PITA because things that would be freaking obvious in person come in dribs and drabs online. Like I just noticed you said "high buzzing comes from the right side of the turntable." What are you talking about? Please be careful to discern between mechanical vibrations you hear emanating from the physical device itself and hum or other noise that is in the electrical signal and so is only heard coming from the speakers.

Ground loop hum is typically about 60Hz, a low throbbing hum, and indicates a ground loop problem. Higher frequency buzzing sounds like your normal run of the mill phono cartridge/phono stage noise. Two similar and related but yet different things. As you can tell.

What I think happened was you disconnect the phono stage this eliminates the ground loop and cartridge completely and all is silent. Connecting the phono stage alone introduces the ground loop. Then connecting the phono adds that buzzing noise.

Now with phono RCA and ground disconnected try the phono stage. If you get hum from the phono stage alone check where everything is plugged in. Try plugging the phono stage into another outlet. Try plugging the preamp into another outlet. If you still get hum like this the last thing to check is disconnect the preamp from everything except the phono stage. If there are no outlets you can plug these into that doesn't result in hum then we have probably ruled out everything and are left with a bad phono stage. 

Otherwise, eventually, if the phono stage is okay then you are bound to find some way of connecting without hum. 

Then connect the phono ground to the phono stage. If this adds hum, well let us hope it does not! Then connect the RCA. Hopefully this only adds the noise. Eliminating this is another similar process. Set that aside for now. Start with the ground loop hum.

Try disconnecting things one at a time. Grounds can be funny. Normally you want to connect the turntable ground to the phono stage. But I went to a new arm where it was the other way around, going silent only with the ground disconnected. 

First I guess start with disconnecting phono stage from preamp. Then work your way back. Connect phono stage only- turntable disconnected from phono stage, phono stage not even plugged in. Then plug in phono stage. Then turn it on. One baby step at a time. Then connect turntable ground. Then R channel. Then L. 

I doubt it is internal arm wiring but if there is no hum until you connect a phono lead then it could be. But check the individual cartridge connections first. I doubt it will go that far. This is just to give you an idea how to track it down.

Everything connected to the same outlet?