Nothing wrong with a calling the flop, if you raise the hand might end quickly (which you probably doesn't want). It's different with two suits, but here I'd think you're at least 90% fav so it's nothing wrong to let some of em catch up a little. I think it's better to build a big pot and take a slim risk of losing it than take down a rather small pot directly.

Turn is fine, river is a raise/call. I think it's more probable BB holds an ace than a king cause the latter he would have bet or raised the turn. If I'm wrong about BB (he actually holds a king) or if poster is slowplaying a king on the turn you lose one more bet. But if poster raise and BB reraises one of them probably have a king and I would fold.
It's a very unusual hand and your longterm result doesn't depend of how you play it (on the river).