I always try to have zero headspace if possible. 3 - 4" is a lot of sq/in of exposed wood that would be a great place for acetic acid bacteria to colonize. I would try and fill the headspace as soon as possible or make sure you have active fermentation so there is always CO2 in the space.
You dont need to do anything different. Adding some fresh yeast at bottling might speed up the carbonation process but in my experience it isnt necessary. I would just add enough sugar to give you 3 vols of carb and call it good.
I recently bottled a Frombiose after a year in the fermenter and another 5 months on wild raspberries.
See here for more - http://alphawolfbrewery.blogspot.com/2013/10/bottling-michigan-lambic-framboise.html
I dont believe it is bacteria but brett forming a pelicle. I always get the same small white floaties after adding fruit to my lambics. It is either wild yeast carried in on the fruit that wasnt already in the beer or the fruit brings a different kind of sugar that causes an increase in that...
I agree with Calder, with the added pack of yeast it fermented to quickly, not letting the pedio and lacto get large populations. It will still sour but take longer. I personally would refrain from adding anything. I wouldnt expect it to be sour now even if you didnt add the added yeast it...
I agree with the others, I have never gotten the amount of complexity and acidity I am looking for with a clean primary and adding bugs in secondary. I would pitch them all in the primary.
I think the problem has to be where your temp probe is located. Try changing it to a T on the output of your kettle and you shouldnt have any issues. This way it is taking the temp of the hottest liquid which is then recirc to the top of your mash.
I really dont see the point in not mashing with your full volume. For me it defeats the purpose of BIAB, ease and simplicity. If you have to heat more water it requires another vessel and another source of heat. For me this is more work and more clean up I have always mashed the full volume...