During fight. The player(number 1) uses a card which given to it an additional throw of a cube for each card dumped from a hand. Together with it spreads on a table of 3 cards which wants to dump. After that player(number 2) cancels a card of player 1 by a spell card. Can the player 1 use one of 3 cards on the table for cancelattion of a spell of the player 2?

No, he can´t, this cards has been used already for the anterior spell

Ok, and can player 1 use this cards to cancel last cards in this fight? If the card of the player 1 is cancelled, whether laid out by it on a table for a cube throw, to it 3 additional cards should return to a hand? Or they should be dumped?

They must be dumped. If the Player 1 discards the 3 cards to pay for the "effect" of card 1, although the effect is cancelled by the card played by Player 2, all costs paid (the played card + all 3 discarded cards) remain so.

Or so I believe.