A USB Microphone as well as any kind of USB Audio device needs a USB host which has drivers for USB Audio devices. Under Win/Mac/Linux these drivers come preinstalled and you can use your USB Audio device in a plug-and-play way. I have experimented with different ways of using USB Audio on Android and this topic seems to be quite diverse.
Depending on the version of Android and especially the Kernel in use you may, or may not, be able to connect USB devices over OTG USB (On-The-Go). With most devices these details don't show in the manufacturer's specifications and has to be tested manually. As far as I got in my research Samsung devices on Stock Android are the most reliable to support USB Audio, but if you start messing with different ROMs and Kernels it is possible deactivate this functionality.
I have been able to record very high quality audio with Urecord and USB Audio Recorder PRO using a stock Galaxy Note 2 and several interfaces (Alesis io4, Shure x2u) and I have been told that Audio Evolution is also able to record from a USB source.
An app that can record video as well as USB audio does not exist as far as I know. And of course, if your device does not support OTG and USB audio there is no way you can use a USB microphone.