Any surge suppressor used on a non ground receptacle(GFCI or not) will likely indicate a fault. These devices rely on the ground to dissipate the surge.
No other device I know of requires the ground. It is only there for safety and the GFCI will provide that function.
When wired this way I assume common testers won’t show the outlet as wired correctly or a GFCI tester won’t trigger it.
Correct. One of those plugin GFCI testers won't work. a home inspector will often call out the GFCI circuit as faulty when there in fact nothing wrong with it.