I wonder if "Any" is equivalent to 1/2/3/4/5/6? If so, then you could place any dice of one "type" at a time.
As for attacking, I understood it that you remove ALL dice when you take the action and the card says "1 hit per die", so it would be all dice result in hits.
This is correct - "Any" means 1/2/3/4/5/6, but you're still limited to placing dice of one type. You could places two one's on one roll, and then one four on another roll, and thus have three dice - 1, 1, 4 - that, if used to attack, would inflict three hits.
You do indeed remove ALL dice on the card every time you take an action or a reaction (so forcing Cromwell to react is a good way to neutralize him scoring all those hits on you).