A nonlegendary creature can’t be your commander, even if it has a “partner with” ability.

The value of X is determined only as Impetuous Protege’s last ability resolves. Once that happens, the value of X won’t change later in the turn even if the greatest power among your opponents’ tapped creatures changes.

“Partner with [name]” represents two abilities. The first is a triggered ability: “When this permanent enters the battlefield, target player may search their library for a card named [name], reveal it, put it into their hand, then shuffle their library.”

Note that the target player searches their library (which may be affected by effects such as that of Stranglehold) and that the card they find is revealed, even though these words aren’t included in the ability’s reminder text.

If the greatest power among tapped creatures your opponents control is negative, X is considered to be 0. The same is true if your opponents control no tapped creatures.