u/Equilibrate321

Hello! I replaced 2 rooms of 2-prong outlets with 3-prong and used a GFCI outlet where the power enters each room. The receptacle tester shows the GFCI works on all of them and that the grounds are OK. But I'm unsure if the grounding is adequate.

The outlets are in metal boxes. I can see the back of the first box in each room. The cable has a ground wire to the box and is crimped to a wire in the cable going to the 2nd box. There is no ground wire inside the box, so the outlet is grounded by the mounting screws. The ground wire is thinner than the other wires, maybe 16AWG. And I can't see how the downstream outlets are grounded without pulling the boxes out.

Is the grounding adequate? Should I label them "no equipment ground"? Did I even need to use a GFCI?