Yes, anything connected to the LOAD terminals of a GFCI device is GFCI protected. So a ground-fault anywhere on the circuit "after" the GFCI device, would cause the device to trip. This will cause the receptacles on the GFCI device, and all the devices attached to the LOAD terminal to lose power.
Connecting GFCI devices on the LOAD side of a GFCI device, should cause no problems other than wasting money.
If you wired three GFCI devices in line as you suggested, and then pressed the test button on the last GFCI in line. The previous GFCI devices should not trip.
Internally, a GFCI looks similar to this.
So when it's set and the test button is pressed, it looks like this.
When the test button is pressed, current flows through the test button, through a resistor, around the current transformer (CT), and back to the grounded (neutral) terminal. I've highlighted the current path in yellow.
So as you can see, you'll have 6 mA on the ungrounded (hot) terminal, and also 6 mA on the grounded (neutral) terminal. This GFCI device tripped, because the return path of the test circuit bypassed the CT. The current will not bypass the CT in the other GFCI devices, so they should not detect the test ground-fault.
This blog post might help you understand how GFCI devices work.