In spite of your obvious efforts to make the questions very clear, it's a bit vague to me (what is "a tabbed document" - do you mean a tabbed tiddler? And I certainly don't understand "use a template to pull in tiddlers") ...but I think I got at least a few things. So FWIW;
For getting a tab to show something else than the title of the tiddler it is showing, you use a caption field in the tiddler.
To get vertical tabs, I'm guessing (based on docs, didn't try) that you just put class="tc-vertical" inside the tabs macro, perhaps at extreme right.
In your macro def, I don't think you should include the currentTddler macro call as an argument. Try to just leav the parameter list empty i.e () and just make the call inside the actual macro where you need it. However, it doesn't look like you're using it so... just skip it.
Hope this is of at least some help. But keep on asking for the other stuff.
<:-)