If anything, I suspected him to be one of the Amyr... If he is a member of the Chandrian, then they operate in an even more strange and inexplicable manner than I previously thought!

Kvothe's mother being Lady Lackless is a good idea, and one that hadn't occurred to me before. It would make sense though, and be properly fitting for the story.

The door in the archives is a total mystery. Whatever is in it, Puppet apparently knows, or at least has some sort of knowledge of it... And I suspect, since Kvothe has yet to be expelled from the University, he will eventually try to break into it. It could contain secret information hidden away, possibly on the Chandrian or the Amyr. Or it could contain some power magic/creatures/evil. After all, we have yet to see any scrael or other such things, so it isn't impossible that the strange room is a type of pandora's box that Kvothe unwittingly opens.

The Lackless box... I have no clue whatsoever. However, I will say that when I was reading it, for some reason I thought of the box that the unhappy boy from Hespe's story used to catch the moon. Very unrealistic, improbable, and likely pointless, but that's what I thought of, and I figure I might as well share it here, since this is all pointless speculation anyway!

And as for Denna's patron, I am thoroughly convinced that it is one of the Chandrian, probably Cinder since he is the one Kvothe hates, so it would be fitting. Her patron is mysterious, enigmatic, likes privacy, does not give away his name, and is overall highly suspicious. He is also sadistic, and in possession of information that Denna wants... All this fits the profile of a Chandrian. However, the question is, what is this information that Denna wants that only he can provide? Whatever it is, I have been thoroughly convinced of Denna's entanglement with the Chandrian ever since at one point in the book Kvothe found a poem about them, and it said something about them coming and going abruptly, and not causing harm (or something like that). And then it cut to a segment about Denna. I don't think this could be coincidence.

I am curious what Cinder was doing riling up bandits in Vintas. It is just a part of their offensive against the entire world, and an attempt to destabilize the countries?

And what role does Auri play in all this? She clearly has an Amyr connection. Perhaps even more importantly, Kvothe gave her a Name. This must have greater significance than simply impressing the Master Namer.

And those are just to name a few. Considering the number of mysteries in the story, the third book will have to be gigantic. Or skip huge periods of time, and leave a great deal unexplained.