Video mode is essentially taking a series of still images, so each image does have a specific shutter speed, and while your mechanical shutter is open all the time, the electronic shutter will still be going on and off. The duration of the shutter speed can affect how "choppy" the final video feels, depending on the amount of motion blur in individual frames. It is also (along with ISO) how you can compensate for brightness from the aperture:
See: What is the "exposure triangle"?
Aperture will always affect brightness--the larger the opening, the more light enters the camera, and the brighter the exposure. Aperture, however, is only one of several factors that determines DoF, and a change in aperture may not actually register as a change in depth of field, depending on those other factors: subject-to-camera distance, subject-to-background distance, and focal length also play a very large role in how much background blur you're going to see. At certain subject distance and focal length combinations, you may not be able to perceive a difference in depth of field with a simple aperture setting change, particularly at the smaller apertures, lower focal lengths, and longer subject distances.
See: How can I maximize the "blurry background, sharp subject" (bokeh) effect?