You right, but this image is actually rendered not by extension but by Chrome implementation of the HTML5 drag & drop. And it was better a few Chrome versions ago...
The fix is possible through... but, it's not a something very urgent, and really require a lot of work to replace this default and automatic behavior.

Actually this soon will be a lot of changes for this edit box (at least i hope).

Ideal behavior already implemented. This text must welcome all new users. It must not be opened in a new window.

It's a real issue, through i must note that this buttons is very rarely just clicked, they mostly dragged i the tree, directly in correct place.

This is mostly a quick hack. One line instead of the day of work and tests... But it must be fixed of course, and replaced by a more polished solution.

About 3) - next major version will actually include shortcuts for this. Without inserting any script in all pages. Chrome recently implement a support for such global actions. This is already implemented in my private build, still cannot manage to release.

Regarding 1) You can create a regular bookmark to Tabs Oultiner window - get a link for it from The Tabs Outliner itself, by right clicking on node that represent its tab ans select "copy link adress". this bookmark will be able to open Tabs Outliner in a regular window

2) as Peter Eliot mention - sometime reading the manual is a good idea ; )

Tabs Outliner create internally (in Chrome profile) several backups of the user tree. They automatically used in case of any problems with main storage (and they happens) but they can also be restored manually. Yet the procedure is not super user friendly.

I record a short screen-cast to illustrate it.

First, dont forget to manually backup your tree using Ctrl-S to HTML file, you then will be needed this file to restore recent nodes.

Then, use developers console to restore some backup:
http://www.youtube.com/watch?v=tpxsxi99CmU&feature...;

In this screencast i first activate a Developers mode in Extensions page, to be able to open Tabs outliner console.

In console you will see a list of files that contain backups of your tree.
Files have 2 numbers in own name, first one is the creation time, in miliseconds, since the 1970.1.1, second one is the number of nodes stored in the file.

to restore any of these backups you need to write in console next command:

localStorage['serviceOptions_restoreSource'] = 'copy-there-backup-file-name.json’

In screen-cast i copy it from other window, using the clipboard, and insert in console, then i select and copy the filename in between quotes.

To force TO for utilizing the selected backup file it must be restarted. So i restart it by disabling and then immediately enabling it. 

The procedure is shown in screencast, i first restore some very old backup with just 5 nodes, and then restore some other.

After that you need to use previously saved in HTML tree to restore the data you have in most recent version of the tree. As the backup will not contain it. Or you can save your older tree, restored from backup, in HTML, then restore some recent backup and then import from HTML deleted nodes back to tree.

This screencast shows how to save tree as HTML and then import some hierarchies back from this HTML file:
http://www.youtube.com/watch?v=j2ckhBNZYhw&feature...;

Hope this will help.

Next version will have much more simple process for all of this, and also a way to see the version of the tree in any previous point of time. But for now this is the only option available.

Best regards.

Well, there is a way, Tabs Outliner store backups for several last days, just in case.
I will describe in few days how to restore the tree from them, don't have a time right now nor even i am close to the PC with Chrome and TO to be able to do so. 

By the way, i highly recommend from time to time expand all nodes and perform Ctrl-S - save as a complete web page. From such files anything can be restored just by drag and drop. For a year already i never delete anything important by mistake yet have several times complete Chrome crash with a complete destruction of all the data in a Chrome profile. This is a much serious and much more real hazard - so having a copy of your tree in HTML form is highly recommended and a must.

Anyway, i tomorrow will get to my computer and describe the procedure of restoring the tree from the internal backup, they are created and stored for last 9 days. You can create a copy now of your Chrome profile, if this incident happened something like 7-8 days ago, so the backup with the data you delete will not be erased till tommorow.