Why the Antonov AN-225 has negative dihedral wings combined with a positive dihedral horizontal stabilizer?
The reason for the wing anhedral has been answered here long ago.

Tail anhedral or dihedral can have several reasons:


To avoid ground contact during rotation (when the aircraft pitches up in order to lift off the ground). This is needed with a swept tail, so tail dihedral can be found on many airliners. Clearly, the An-225 tail has ample distance to the ground, because the vertical tail surfaces extend for ⅓ of their span below the horizontal tail, so any ground contact can be ruled out even for an un-dihedraled tail.
To move the horizontal tail away from the wing wake. When you look at the tail of the An-124, you will see no dihedral and a mounting point quite similar to that of the An-225. But if the An-124 has no problem with an un-dihedraled tail, so should the An-225. This can also not be the reason.
To add some more vertical tail surface. This trick has been employed on the Handley-Page Victor or the F-4 Phantom II. On the An-225, the outer verticals would shield much of the horizontal tail and little additional side force will be created by the dihedraled horizontal tail. So this cannot be the reason, either.
For a H-tail: To move the attachment point of the vertical surfaces closer to their middle. Imagine the An-225 horizontal tail had no dihedral: Then the verticals would need to be mounted on the tips of the horizontal near their bottom (they need to be so high up in order to allow the An-225 to rotate, after all). This would mean that all side loads would act above the attachment point, producing a sizeable bending moment which has to be carried along the spar of the horizontal tail, making the outer tail structure a lot heavier. With a little dihedral, the attachment point can be moved further up, close to the middle, so the bending moments in the vertical and horizontal tails can be a lot lower.


Antonov has designed H-tails before (on the An-14, the An-22 Antei, the An-28 and the An-38), and in all cases the vertical had a sizeable part of its surface below the horizontal tail. The only design on this list which also had some tail dihedral is the An-14 from 1958, and here the attachment point is almost in the center of the vertical. So it looks like the Antonov design bureau applied a trick on the An-225 which they had known for at least 30 years before the An-225 took off for the first time.