You can activate Maddening Imp's activated ability "only during an opponent's turn and only before combat" (C.R. 108.1). If you attempt to shorten the shortcut of the opponent moving to the end step by activating that ability, you have to propose to activate it during that player's first main phase or earlier, if possible (C.R. 500.2, 720.2a-c, 505.1; see also C.R. 506.6d; for sanctioned tournaments, see M.T.R. 4.2). If you do, your opponent can now choose only to shorten that shortcut or accept it (C.R. 720.2b; for sanctioned tournaments, see M.T.R. 4.2).
The Imp's ability can only be used precombat on the opponents turn. So assuming they are in their precombat main phase and declare they want to end the turn what that are actually proposing is a short cut in which all players will pass priority until their end step. As with any proposed short cut you can interrupt the short cut at any point. For the bat this will be still in precombat main phase. The ability will be put on the stack and the turn will continue as normal. (Interrrupted short cuts don't continue they are ended where you interrupt them)
As for the second question. No the opponent can't end their turn in response all movements of the game require input from both players. Unless of course they have Sundial of the Infinite.
A player can not unilaterally "announce EoT". They can suggest a shortcut in which both players do nothing but pass priority for the remainder of the turn, and they can start that by passing on their next priority, but that would merely expedite the resolution of something that is already on the stack.
720.2. Taking a shortcut follows the following procedure.
720.2a At any point in the game, the player with priority may suggest a shortcut by describing a sequence of game choices, for all players, that may be legally taken based on the current game state and the predictable results of the sequence of choices. This sequence may be a non-repetitive series of choices, a loop that repeats a specified number of times, multiple loops, or nested loops, and may even cross multiple turns. It can’t include conditional actions, where the outcome of a game event determines the next action a player takes. The ending point of this sequence must be a place where a player has priority, though it need not be the player proposing the shortcut.
720.2b Each other player, in turn order starting after the player who suggested the shortcut, may either accept the proposed sequence, or shorten it by naming a place where they will make a game choice that’s different than what’s been proposed. (The player doesn’t need to specify at this time what the new choice will be.) This place becomes the new ending point of the proposed sequence.
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What happens when I activate Imp's ability in response to opponent ending his turn?
Can opponent respond to the Imp by announcing end of his turn?
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As for the second question. No the opponent can't end their turn in response all movements of the game require input from both players. Unless of course they have Sundial of the Infinite.
Shu Yun, the Silent Tempest WUR Voltron Control
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A player can not unilaterally "announce EoT". They can suggest a shortcut in which both players do nothing but pass priority for the remainder of the turn, and they can start that by passing on their next priority, but that would merely expedite the resolution of something that is already on the stack.