Re: TOC on the top
I see your points: having the title of the currently displayed topic hovering on the top is indeed nice. And expanding this same line into a TOC also feels right. But the combination of the two means that you can see either the TOC or the board, but not both. And that feels very wrong to me. I'd prefer having a similar, a bit more compact version of the whole thing floating on the right side.
This might be to some extent a matter of 'taste': just like some people prefer step-by-step driving instructions while others prefer a route indicated on a map, there's also different ways of navigating a web page that work best for different people. So the answer could be to make it configurable. By user preference, skin, custom css or - my preferred option - a switch on the UI, comparable to the show/hide switch on article TOCs. One could then maybe collect usage data to see how popular each display style is, and how frequently people switch.

Re: Whitespace on the right
A nice start would be to change the behavior for small browser widths: when reducing the width, the first step should be to gradually reduce the whitespace all the way to zero, then reduce the board width, and then the font size. Currently there seems to be some kind of requirement of a certain minimum amount of whitespace, which is only dropped once the font size is reduced.
That said, the current behavior is exactly what we wanted if there was something useful in that space (hint, hint ;-)

Re: Showing similar data in regular TOC
+1

Re: Will < > be used?
Definitely, at least the ">". That's what people do with the browser's search function on non-infinite pages. It would be great to map this function to Ctrl-G. (Actually, this might be a 'must have': otherwise Ctrl-G will repeat the latest native browser search within the loaded content, which is confusing if you did another search in between.)

Re: Infinite scrolling / loading topics
In the prototype, topics without search results are hidden from the toc, but still shown on the board. Will that be different in the real version (the board then showing only topics with hits)? I think it should.
I agree that the scrolling-up issue is independent of searching or other filters, but there's also a potential search-specific problem here: the set of topics to be shown can change drastically with every keystroke in the search bar. Maybe the reduction of the board to showing only topics with results should only happen when pressing enter in the bar?