BadUsb works by infecting a computer when plugged into the USB port. Is it also possible for an attacker who gains control of a system, such as remote access, to then infect any connected USB devices? Or does it only affect the ports it's plugged into?
Once an attacker gains control of a system he may use it to try and compromise vulnerable USB devices plugged into it in hopes of compromising the other systems to which the compromised USB devices will be connected.
And the attacker doesn't care about which ports are used, he'll just probe all of them in hopes of finding & compromising a vulnerable USB device.
If a USB stick accepts unsigned firmware updates then yes it's vulnerable to a bad USB attack. There are products out there that only accept digitally signed firmware updates. These devices are not susceptible. http://www.ironkey.com/en-US/solutions/protect-against-badusb.html