Meantime you can export session to html file, through Ctrl-S.
This is not just a hack but a specially supported mode, so you can drag hierarchies back to TO from this file.
Regarding other methods of export/import and synchronize - they are also planed, but i prefer not to promise any exact dates of when.

It's possible to assign a keyboard shortcut to open TO (in the Chrome Settings\Extensions), but i think it will open TO only if the Chrome already opened and focused.

As for opening TO by some icon in task bar... well, as you already found it's possible to open it in a regular tab. Opening it directly as stand alone application is planed, but not in the near future i think...  yet I will think about adding the option to always open TO in a panel like window, this will close this case.

Yep, need to check them. Will do this after the next major release, it is anyway hugely rewritten so there is no sense to check with the current version. Meantime the  next major release  of TO have more priority then the Opera or Iron support. 

I do plan support the Opera actually.

But if for true i am not very exiting about this idea, it's hard enough already to support the Chromium and retest on all of the combinations of OSes and browsers.

First of all note that you can move the whole windows hierarchies, by drag and drop, from the saved HTML file to the TO window.

There is also possible to transfer the whole database to a new Chrome instance:
https://groups.google.com/d/msg/tabs-outliner-support-group/eKubL9Iw230/oLZlzTUb7R0J
But please make a Ctrl-S backup before this, as during this you will lost the present data in the target instance of Tabs Outliner

It's starving without funding.
As many other important things.

Through soon there was a summer vacation and i hope to make some steps forward.

Will be cool if you list there this many ways to generate money after given away the sources. Please when you will do so consider the project about we talk. It's not operation system, not developers library, not a browser under the trans-galactic corporation (with millions other paid services) shoulders, not a soft for some device that is sold separately.

There is also many similar tools in a same field, direct competitors, with near half a million users count, who will be more then happy to take the sources, and just publish them as updated version of their own tool.