Unusual Receptacle Wiring

Hi. I have a house built in 1970.

I was going to replace the receptacles in one room which are all middle run when I found two things to be unusual.

First, the ground wires from each cable are connected to the two grounding screws on the metal receptacle box, one continues to the receptacle grounding screw and the other is cut short at the second receptacle box grounding screw. They are not pigtailed and are not connected to each other and seem to use the receptacle box for continuity.

The other thing I found unusual was the way the receptacle was wired. Normally the black and white of the first cable are connected to the top receptacle screws and the second cable, the black and white are connected to the bottom. The receptacles in this room are somewhat reversed. The black wire of the first cable is connected to the top screw and the white to the bottom. The black wire of the second cable is connected to the bottom screw is the white connected to the top.

My DIY receptacle checker tells me all is well.

My questions are:
- Should I bother trying to pigtail the ground wires and re wire them properly or is it alright as is?
- Should I bother rewiring the receptacle wires or leave as is?

Any comments or insights in this and why my receptacle checker tells me all is well would be much appreciated. Thanks.

Avanzi