Youtube – How to remove a video from the currently playing playlist

Try EndlessYoutube. Obviously it will still only work for videos that allow embedding but it should be less annoying.

The playlist is available in the current design (One Channel, which you're referring to as "new design") if the user has featured a "newest video" shelf on their channel page – click the "play all" button.

In the now-new design (Material Design), there is a link to play all videos on the /videos tab of any channel.