Video recording is not currently possible in Inquisit Web (audio recording is, though the recording time is limited).
> In addition, when we screen participants’ recording quality before the start of the study, is it possible to connect the recorded audio/video file to another server, such as MATLAB, so that we can run our own code on MATLAB to examine the sound quality and determine in real-time whether they can move onto the experiment?
No. Inquisit does not stream audio back; any recorded audio is stored locally on the client device and the recording is only transferred to the server when the experiment is finished, along with all other data collected.
> Also, can we do a multi-player study where paired participants can interact through audio or video on Inquisit 6?
You may be able to have your participants interact via a different app / channel (e.g. Zoom or Skype), though, while they complete some Inquisit task.
A multi-player study in the sense that the Inquisit task running on participant A's device reacts to something that participant B does in the Inquisit task on their device is not possible -- there is no way to connect these instances over the web.