USB Blaster and other JTAG devices in chain

The question goes to the other device's vendors, I think. Besides initially reading all device ID registers in the chain, I would expect Quartus to bypass the other devices. JTAG is designed this way. You also shouldn't need to use TRST to operate the JTAG interface.

But you can try to pull TRST low to prevent the other devices from acting on misunderstood JTAG commands. If TRST is floating, it may be actually a reason for the unexpected behaviour.

To operate Quartus tools with third party devices in the JTAG chain, their BSDL files must be known to Quartus.