Can "aggressive" or unusual play in blackjack get someone thrown out of a casino?
I think winning an excessive number of times get you thrown out of a casino. And the casino will then try to find "good" reasons like card counting or whatever dodgy moves you made. If splitting tens is one of them, and you placed a high bet and you won a lot, then certainly the casino will use that to support their claim. If you just split tens and then lose, I don't think the casino will throw you out. :)

I think the unwritten rules that you mention are a set of heuristics that the casino uses to decide whether or not you are suspicious. More or less like a spamfilter deciding whether an email message is spam or not. Some actions make you more suspicious and once you go over a threshold the alarm bells start ringing. Of course the number one alarm bell would be the fact that you are winning, but I can imagine that a persistent ten-splitter might eventually become suspicious for being "out-of-the-ordinary". I just don't think the casino will chuck you out as long as you don't win too much.

Regarding your card counter's book. In the end (s)he is a card counter, so if the casino chucked them out for splitting tens perhaps (s)he had been doing other moves in line with being a card counter inherently, and has, perhaps without knowing, made several other alarm bells ring already. Maybe the splitting of the tens was "the straw that broke the camel's back"

As far as my knowledge reaches regarding Blackjack, I believe in order to split tens they need to be the same denomination. So a Jack and a King cannot be split.