Well, an RPG has you play the role of the character by giving it stats to represent how good that character is at different things (instead of it being based on player skill). So yes, every RPG has role playing, just not the type you're used to thinking of.
What? No. You aren't playing that role, the character is playing that role. That would be like saying watching TV is role playing, since the character you are watching is playing his or her role correctly. All you're doing in most RPGs is moving the guy around, but his stats and dialogue are chosen by the game, so the game is playing the role. I hope that makes sense to you!
In Half-Life 2, for instance, the game forces you through a very linear path, but you're still playing the role of Gordon Freeman. You are given a physics puzzle, and the only way to solve it is to use the gravity gun to move objects around. That's Gordon's defined role, and you are playing through it. You're not really given a choice, but you're filling his shoes.
Most RPGs expect the player to go on adventures and, in that regard, that is the role you are playing: adventurer. That's about as deep as it goes, though. Most CRPGs are different because they allow you to select stats and which type of character you build and how you tackle missions or which adventures you take on. There is a higher level of control in most American games than there is in Japanese games; exceptions naturally exist, but for the most part American RPG designers give the player the ability to control their actions and, ultimately, impact their destiny directly, whereas Japanese RPGs (especially lately) have been becoming less and less interactive stories where you are really just moving the main characters from cutscene to cutscene. Again, you're not playing a role, you're just... flipping pages.