As a hotfix for 1, you can use the Sheep icon (clone) to create another Outliner window that does act like a normal tab. Unfortunately the extension's main icon button will still always open the thin window.

For 2, you can save/open whole groups in the tree as long as you collapse it first.

For 3, you can Save tabs /if they're in a window-node with saved items, and you close the whole window at once/. But yes, I agree that a way to save individual tabs from the page itself would be nice. Vladyslav has said a keyboard shortcut for that is unlikely because it requires inserting javascript into every page, but that he might do something with option in the icon-button to do that.

A shortcut I would LOVE to have is a way to close a tab/window from the tab/window itself, while saving it. Having to find it in the Outliner window is a pain when I just want to save-close it. Something along the lines of cmd-alt-w for save-close tab, and cmd-shift-alt-w for save-close window.