When I'm traversing deck in Search action should other players see the cards that I'm checking?
Yes, those cards are "flipped up and revealed" (see TBoW page 6).
Again, when some player checks some number of cards using consume power of Gambling Word should other player see the cards he is checking?
Yes, those cards are "flipped over" (see RvI, page 10).
But should other players see what card did he choose as a reward if he won?
Yes, he chooses "any one of the cards flipped over" (see RvI, page 10).
So cards consuming UP TO some number of cards (at least 2) like Old Earth or Consumer Markets also match this [Consume power Search] criteria?Yes, these cards have Consume powers that "can consume two or more goods".