How fast would an object at cruising altitude(around 10,600 m) need to travel to stay on the opposite side of the Earth to the Moon? I am trying to figure out how fast an object will have to travel (at a height around 10,600 m) to never be in the presence of the moon. Meaning the object will always be on the opposite side of the moon to the Earth.
Well, as a first approximation,  you know the Earth rotates once every 24 hours while the moon moves "only a little" in that time.  This means your object must appear to an observer on the Earth' surface to rise and fall 12 hours opposite to the moon.  First thing, then, is you better be moving East to West, just as the moon appears to do. 

Using the Earth's radius as approximately 6400 km,  your object is essentially at the Earth's surface to 2 significant digits.  The orbital circumference is thus $2*\pi*6400$ km.  You want to cover that distance in  24 hours, or 1676 km/hr . 

If you want to get very accurate results, you'll need to take into account the moon's eccentricity with respect to the equator (so that you are always on the opposite end of a diameter thru the Earth's center from the moon) as well as the fact that it moves approximately 1/28 of a revolution per day.