AFAIK wireless cards don't have firmware ROMs, which means that the firmware is loaded at hw initialization from your hard drive.
[ 125.610092] phy0 -> rt2x00lib_request_firmware: Info - Firmware detected - version: 1.7.
. You just find the firmware file, and put it in /var/lib/firmware. It does nothing for windows, as windows has its own copy of the firmware (which you also could try to use, if you find it).
However, firmware updates aren't too common, I'd suggest trying to update your wireless drivers instead. The problems with the intel cards are a bit weird, because those are pretty old models, which I know work on WEP/WPA networks. Isn't the problem that your network is WPA2? I'd look into the configuration differences between those networks if I were you.
btw : what distro are you using, kernel version?