Just for a reminder to include this info:

Copyright is generally a dark art and full of exceptions, and action can often be down to the mood and whims of the creator.

When considering videos and screengrabs and gifs, there is kind of a 'common sense' approach when you read the terms on YouTube and I think this should be followed here too.

When one makes a video that responds to another video, they can use a certain amount, within reason, of that original video without harm. But as an aside to that, does their actions actively harm the original video's user, does it take away income by bringing users to the response video instead of the original?

There's a lot of things to consider that often end up on a case-by-case basis.

Personally, screengrabs, as long as they are properly credited, are fine, since they only represent one frame. But if the screengrab is of an image that is broadcast onto the video, then the same rules apply to any regular image.

Movie screengrabs are fine, if credited correctly, and hand-drawing a logo is generally fine as long as it iisn't obviously trying to complely mimic the originial. This often depends on the copyright standards set by the artist. Sometimes they say their image may not be replicated or modified, others say this is ok. It depends on the licenses and its your job to read up on that.