( A LDO is basically a high gain amplifier, so it oscillates great if you get the wrong output / input conditions )
Yes but an 7805 is not a LDO regulator.
unlikely from what you say to be a capacitor problem, that is noramly an oscilation problem.
That can cause things to shut down.
ok...I really need help now.
Yes and it is your own hands. Your description of what you are doing is less than clear. It is full of generalizations so it is difficult to spot any errors you have made.
Your schematics have been quite poor and do not show what you are wiring to what.
First thing to do is to find out why your power supply shut down. The normal reason is due to over current but it could be that you are feeding some voltage into the system or you power supply is not a floating one like you thought.
Is it likely that somehow the ground is sending some unwanted noise
No.
It does not help having a mystery 'downstream device' in the mix.
And do you think it's necessary to add another capacitor between 7805 ground and -28V??
Yes, the 7805 must have a capacitor on both input and output, it says so in the data sheet so do it. Only an idiot ignores what it says in the data sheet.