Round penalties are computed in accordance with TDA rule #40, which says in pertinent part "...The offender will miss one hand for every player, including the offender, who is at the table when the penalty is given multiplied by the number of rounds specified in the penalty..."
So, if there are 10 players at the table and the player holding the button is assessed a one round penalty that starts with the next hand (player is now the Cut-Off), the first penalty hand would be the player to the immediate left of the penalized player (who is now the button). So you just have to count the number of hands, lets assume it is a one round penalty. The first hand is when the penalized player is the Cut-Off. The last hand of the penalty would be the hand in which he is the Button. He is allowed to come back to the table when he is again in the Cut-Off position.
As to your last question it is not material since he cannot play the Button. But if he could, he could come back on the Button because for the entire penalty period, he is dealt cards and MUST pay all Blinds and Antes. It is no different than it would be if the player was in the restroom or at the bar. In a tournament that player is dealt cards and has to play the blinds, etc.