Does distance matter for gravitational U in non-earth/ non-celestial-body systems? Gravitational U = -Gm1m2/r, thus the distance (r) the two carts are apart affects how much U you end up with, no? Thus it also affects whether or no ME is ultimately conserved, because you can just set the distance so that it is. Can you even freely set the distance/ where h=0 is, even though it is not an earth-object system? So because of such variation, I feel like distance does matter in these cases, and you need to be given the distance between the objects to answer the question. 

Consider the time interval from when the two carts are released until just after they collide. For the system consisting of only the two carts, indicate whether the total mechanical energy increases, decreases, or remains the same. (The carts start out at rest btw) 

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 A: Mechanical energy is the sum of potential energy and kinetic energy.
Potential energy is the gravitational potential each cart has due to being raised distances $2H$ and $H$ from the surface.
This is not referring to the very very small amount of potential energy they would have if considered to be mini planets at a distance $r$ apart, as your equation suggests. This "celestial-body potential energy" is negligible here as they have such small masses so we do not take into account when working with mechanical energy.
So, yes, you are right that for a true proof of conservation of all types of energy you would need to be told the distance apart that they started. This would allow you to calculate the loss in cart-to-cart gravitational potential energy lost due to work done by the very small gravitational force of attraction between the carts. However, all questions of this sort are presented in an abstract framework where things are considered to be "ideal" and negligible effects, like this tiny potential energy, are ignored.
