Tube PHONO preamp interference - RFI, EMI, bad grounding?

@ejb14 It's connected, both channels are. The grounding pin of the power input is connected, the case as well (but's it's painted so it can only be measured through the screws or one tiny spot where the paint has been scratched off)

How's that possible the shorting plugs aren't working? (This pictures shows the signal wire and the groung wire of that RCA plug before I twisted them together)

Is it possible that the tubes pick up the signal before it gets grounded? 

@audphile1 of course not. Not even plugged in. But I powered it on when I changed the tubes and left the cage opened - to see if they all light up. 

I think your shorting plugs are doing their job, and you have verified that your input shields are properly connected to ground and that the problem lies elsewhere.

Have you verified your power outlets (in the wall) have good ground? You can get a tester cheap at a hardware store. That is another difference between your place and where the EAR worked just fine. 

FYI, I just found this link and this guy details a lot of the steps you have done so far, including the shorting plugs. Worth a read - 

Trouble shooting phono amplifier and SUT noise

@filipm 8 days later and you are exactly where you started. You have a broken or poorly designed component or the combination of the two.
You should reach out to the person you bought it from and start working thru some sort of resolution. If it’s too late for that, may be send it in to have a qualified repair tech have a look at it and determine next steps. 
Doesn’t look like you’re on the right track here…

@filipm I am not convinced you have a broken or poorly designed phono stage. 

Recap of the evidence so far:

1. You have a predominant 50hz hum in the noise as was analyzed previously.

2. Both the EAR and the ARC LS16 exhibit this noise in your apartment

3. Both the EAR and the ARC have three prong power inlets

4. The EAR routs its ground through the third pin of the power inlet as confirmed by your connectivity test. Presumably the ARC does as well. My Herron tube based preamp, and my Bottlehead tube headphone amp do this as well. 

5. The EAR does not exhibit the noise when used in another location (when plugged into a power outlet not in your apartment)

6. The shorting plugs did not reduce the noise

7. Grabbing/touching the cables affected the noise

8. The troubleshooting guide I linked, Bottlehead's trouble shooting guide, and @hagtech 's guide all suggest a 50/60 hz hum/noise is a lifted ground

This evidence suggests you have a lifted ground, possibly either in your power strip, or in your apartment outlets. Assuming the LS16 is not broken or poorly designed, and exhibits the noise too suggests that the EAR may not be broken. That would not make your EAR broken or poorly designed, just not being used as specified (with a good connection to ground).

Assuming the OP lives in the US, I think it's a bit too simple to blame his issue purely on bad grounding, if you listen to the recorded noise up above.  First of all, there is an intermittent burst of 50Hz noise.  In the US, we use 60Hz AC, and it's pretty well regulated at that frequency. Typical noise due to grounding issues is at 60Hz or 120Hz.  Moreover, it's not intermittent; it's constant.  This noise seems due to RFI or EMI from some outside source, though it's true that the susceptibility to the Interference could be related to a poor ground circuit somewhere.  There are already a myriad of reasonable ideas for the source and what to do about it in this thread.  I hope something works.  I am now wondering about bad diodes inside the EAR or the ARC.  I am also wondering what did happen or will happen when all the tubes are replaced, though that will not rule out a cold solder joint inside one or the other device.