If there's a decently sized moon (say, 50% visibility or more) but you still have a few hours of time before moonrise/after moonset, will the moon still wash out stars the same as if it were visible in the sky?
The strict answer to your question is no. I say strict because your question says "the same as". The degree of wash out is related to proximity to the moon and, of course, the brightness of the moon. When the moon is visible in the sky the area of wash out is greater than when it's below the horizon. The further away you look from the moon the less the wash out. You'll get some wash out at the horizon where the moon is even when the moon is below the horizon. This will decrease as you look further away across the sky from the moon's position. The spread of the wash out will also depend on the atmosphere. If you are in a murky, polluted environment there'll be more Rayleigh scattering of the moon light and the wash out effect will be greater.
I had exactly this experience last night when I was making a time lapse of Orion hoping to catch comet Lovejoy. Even though the moon was below the horizon for the whole sequence I could see the wash out area increasing with time.