No, you may not download, reproduce, or screenshare YouTube videos. When using YouTube you are subject to its terms of service, which specify that you are not allowed to download or redistribute content on its platform. The Permissions and Restrictions section of YouTube's terms of service (last updated on January 5, 2022) includes several restrictions to user who are not permitted to "access, reproduce, download, distribute, transmit, broadcast, display, sell, license, alter, modify or otherwise use any part of the Service or any Content except: (a) as expressly authorized by the Service; or (b) with prior written permission from YouTube and, if applicable, the respective rights holders." Moreover, users are not allowed to "circumvent, disable, fraudulently engage with, or otherwise interfere with any part of the Service (or attempt to do any of these things), including security-related features or features that (a) prevent or restrict the copying or other use of Content or (b) limit the use of the Service or Content."
If you want to use a YouTube video in a way that is beyond these restrictions, we recommend that you reach out to the rights holder to ask for permission and for access to a copy you can use.
For more information on linking to online videos, see question 22: May I include links to videos from a free platform such as YouTube or Vimeo, or embed videos on a learning management system such as LEARN?
For more information on how to ask permission, see question 14: How do I get permission to use someone else’s work?
Note: Quotations from YouTube's terms of service are from the version that was last updated on January 5, 2022, and these conditions are subject to change. Review the terms of service directly for more up-to-date information.