If I used Take the Black on, say, Tumblestone Knight equipped with Ice.1) As you said, when you take control of a card, you only take control of *that* card - not any peripheral entities, like attachments, duplicates, etc.
Could I sacrifice Ice in a millitary challenge that I win?
As far as I know the Ice is still under my opponent's control, but it's sounds kinda silly if I would get +2 but couldn't sacrifice it.
Or else, if it's still under opponent's control - could he sacrifice if I'd lose a challenge in which the one with Ice is participating under my control?
2) You can only initiate abilities on cards you control. Since you do not control Ice, you cannot initiate its ability. However, the constant STR modifier still applies (just like Milk of the Poppy still applies!)
3)Yup. The card text is always read from the point of view of the controller, so you understand it correctly. If your opponent wins a Mil challenge in which (now) your Tumblestone Knight with his Ice is participating, he could sacrifice Ice to kill a character you control. Like the Tumblestone Knight. Wasn't fit to weild Ice!
This sounds really odd, but but if you're stuck with a bunch of high STR No-attachment characters, who knows, it might be a good idea to play Ice on your opponent's only character with a mil icon as a way to get an unopposed challenge. Definitely a weird/fringe play.