Can link previews be disabled?
Yes. Sending link previews is completely optional. You can disable or enable this feature anytime on your mobile device and the change is synced to your linked devices:

CoolApp Settings  > Chats > Generate link previews

You can also remove any individual link preview by tapping on the X icon in the corner of the preview before sending a message.

Which sites are currently supported?

Link previews are available for all sites that explicitly start with https.