The Bridge is in essence a wireless access point in reverse. it attaches to the wireless signal and transfers the network stream to the wired ethernet jacks. The benefit is you have have "wireless" to things that wont accept a wireless module. The wireless network adapter for the original xbox was a wireless bridge, and will work exactly the same. they just charge more for it being official.
A standard usb wireless should sync with any standard wireless router or access point. The problem in using them for consoles is the console will usually not recognize the usb device as it has no drivers, and no way to install any. the usb wireless modules that they sell for a certain console are designed with the console in mind so that it knows to recognize and use it.
the "gaming routers" work just the same as regular routers, but they usually have an increase in processor speed or memory inside the device, giving it more power to process data faster. does this make for less lag in games? yes. Is it worth the extra $$$ ? Depends on if a regular router gives you a bad experience in games. But in my opinion your going to have more lag from your internet connection then what your router is going to add.