Will GFCI outlet protect downstream outlets with ground connected to neutral?
Never never NEVER connect the ground of an outlet to neutral!!! Depending on the situation, you can end up with a dangerous or even lethal voltage on the ground and therefore, on the case of something plugged in.

It is legal and advised to use a GFCI to install 3-prong outlets on an ungrounded circuit. This also applies to standard outlets downstream of a GFCI. The outlets must be labeled "GFCI protected, no ground" (labels should have been provided along with the GFCI).

You should remove all of the "bootleg" grounds.

BTW, the reason the tester didn't trip the GFCI is that it works by connecting line to ground through a resistor to simulate current leaking to ground. However, because the outlet ground is actually connected to neutral, there is no actual leakage.