How are Max Brake Energy Speed & Weight affected by other factors?
The factors you mention will affect how much energy the aircraft has that's going to be transferred into the brakes, and how much will be lost otherwise.  The more energy you have, the slower you can go before the brakes are overwhelmed -- thus, reducing your VMBE.

With more flaps, energy will be dissipated in aerodynamic drag, although probably not all that much (though going from the least possible to the greatest possible flaps may make some difference -- or it may be sufficiently slight that the charts are published with a worst-case scenario &amp; if you have greater flaps then your situation will be slightly better than what's charted).  With runway gradient, you'll dissipate some energy rolling uphill, while you'll have more energy (potential energy becomes kinetic energy) rolling downhill.

The really significant factor is aircraft weight, as your formula makes clear... an aircraft stopping at 100,000# will have 2/3 of the energy of an aircraft stopping at 150,000#, all else being equal.  So, in those two cases VMBE would be much higher for the 100k case... if Limit = 0.5 * wt * V-squared, you can have a higher V-squared to reach the same limit with a lower weight.

Contamination probably wouldn't matter much: while you would be transferring energy more slowly into the brakes stopping on a slick runway than on a dry one, the total energy absorbed into the brakes would be the same.  So unless the rate that the energy is absorbed changes the maximum energy that the brakes can absorb (seems like it would be unlikely, although in some cases... who knows), contamination wouldn't affect VMBE.  (Unless the charts take stopping credit for the deceleration due to "plowing through" contamination -- which would actually help out to some extent.

The most likely effect would be to make the stopping distance more limiting in the contaminated case than it would be otherwise -- so you might be limited by the ability to stop before the end of the runway in a contaminated case, where you'd be limited by brake energy in a dry runway case.

The effect of weight, flaps, and slope on the actual VMBE would be that things which give the aircraft more energy (more weight, less flaps, downhill slope) would reduce the VMBE.  Think of it in an extreme case... if you had the brakes of a bicycle, a one-pound vehicle could go insanely fast before you could put enough energy into the brakes to cause a problem, but if your vehicle were, say, the Space Shuttle, even at slight speed, the brakes would be overwhelmed by the KE of that much mass.  Consider likewise, a 45 degree downslope vs a 45 degree upslope, or the drag caused by a huge drag chute vs the drag of a pencil flying through the air.