Do 5 connected pawns usually win against a rook? Suppose, white has 5 connected pawns, black only a rook. Under normal circumstances
 (black cannot immediately win a pawn, the white pawns are not too far advanced, say
 on the rows 2 to 4, black's king is standing in front of the white pawns)

Can the pawns triumph in such a situation ?
Yes of course, there are more chances for white who has 5 connected pawns to wins instead of black who has a rook. A rook can't check the king all the time and the pawns are always helping each others to get to the final state to get a queen. This problem is usually won by the 5 pawns.