Will outlet tests downstream of a GFCI outlet show open ground?
The stickers are mandatory.  If you don't like the aesthetic of the ones that come in the GFCI box, feel free to &quot;brew your own&quot;. They only need to say the words.  And while you're at it, have them state where the reset is located.
If you properly install 3-prong receptacles without ground after a GFCI device, you should expect that a 3-light tester will read up 1 yellow light only (no second yellow). If you read the sticker that interprets &quot;1 yellow&quot;, that will suggest &quot;open ground'... but all those interpretations are useless, misleading and a waste of your time. Ignore them.
If you push the &quot;GFCI test&quot; button, it will be normal for it to fail to trip the GFCI.
Keep in mind that novices are poorly equipped to recognize two exceptional situations:

No ground wires, but there is metal conduit or metal jacketed cable which is an approved grounding path.  It is normal not to see ground wires in this case.
There is flex metal conduit or metal jacketed cable which is not an approved grounding path, but does connect ground well enough to operate the 2nd yellow light, to trip the GFCI, and provide transient ESD protection to equipment.

An example of unapproved grounding paths is BX cable, which has a spiral shield with poor electrical conductivity across the spirals. So it isn't like a wire of its width, more like a much longer spiral ribbon resistor.  Good enough for ESD and GFCI... but if a dead hot-ground short occurs, a common condition, it will not flow the 100-200A needed to get an instant trip from the breaker.  Instead it will &quot;glow cherry red&quot;, potentially starting a fire.