Is it permissible for me to download videos from YouTube onto a private forum of mine, on the condition that only I and my friends watch them, without any indecent content, and with the YouTube link remaining?

It is permissible to benefit from YouTube if one can avoid temptations. Transferring beneficial videos from YouTube to a forum is a good deed that reduces exposure to YouTube's corrupting influences. Transferring videos does not violate anyone's rights because YouTube's content is available to everyone.