- or I am not allowed to play my unique card (in this case until what: till the opponent's card under my control is dead?)?

You cannot play your copy of the unique card until the opponent's card is no longer under your control.

- In order to play a unique card, you may not have another copy that you own or control in play or in your dead pile.

- In order to play a dupe, you must own and control both the copy you have in play and the copy you want to dupe it with.

So, simply controlling your opponent's copy stops you from playing yours ("or") in any way. You have to wait until you don't control the opponent's copy anymore before you can play yours. That loss of control could come because the character dies (and goes to your opponent's dead pile), but it could also come from leaving play some other way or your opponent taking control back.

Well, thank you, ktom!!

But then appears another question ) May I get a control over opponent's unique card if I already have such a card in play?

The FAQ specifically addresses this in section 3.29

(3.29) Unique Cards and Changing Control

You may not play, put into play, or take control of a unique card already in play that you own or control (except for putting a duplicate on a card that you own and control), or that is in your dead pile.

Nope. The rules for unique cards say that you cannot "play, put into play, or take control of a unique card already in play that you own or control" or that is in your dead pile.

Ah, I've missed that in the FAQ - thank you both!