Re: strange passive donk check
The BB's combined flop and turn play is very typical among semi-aggressive mid limit players who have some sort of weak made hand (i.e. 2nd pair) + draw, that missed on the turn. You are almost certainly good and should bet. There are some players who will checkraise their draw twice - you get to know them, and you just beg them to try it on this board. But I don't know your opponent so I can't tell you what to do if checkraised.
Since you didn't ask, I won't say that overlimping a bad offsuit king in MP is generally a -EV play.
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