Does antimatter curve spacetime in the opposite direction as matter?
Antimatter has the same mass as normal matter, and its interaction with gravity should be the same according to GR and QM.
That said, antimatter has only been created in tiny amounts so far and only few experiments have been performed to confirm there is no new physics involved.
The gravitational interaction of antimatter with matter or antimatter has not been conclusively observed by physicists. While the overwhelming consensus among physicists is that antimatter will attract both matter and antimatter at the same rate that matter attracts matter, there is a strong desire to confirm this experimentally, since the hypothesis is still open to falsification.
https://en.wikipedia.org/wiki/Gravitational_interaction_of_antimatter
See also: What is anti-matter?
Currently there is no reason to believe/require antimatter has negative mass. It should therefore behave exactly the same in a gravitational field.
The matter-antimatter distinction is pretty arbitrary. We found protons/neutrons/electrons first, so particles of the same families that exhibit similar behavior are "matter", and those with certain properties (charge, baryon number, or something else, depending on the family) as opposite would be antimatter. We could call positrons as matter and electrons as antimatter and nothing would change except for our definition of lepton number (and the labels of the muon/tau).
When Dirac calls it a negative energy solution, he's looking at the case where we have a sea of ground state matter, and we excite one. The "hole" left behind by the excited particle behaves like the particle itself, but can recombine with an excited particle with no net energy change so one can view it as having a negative energy.
In this case, the hole does have negative mass because it is in a "sea" of positive-massed particles, and removing these leads to a hole with negative mass. And it behaves similarly from the POV as gravity.
In the general case, an antiparticle has the same energy as a particle.
Here's a naive argument for expecting antimatter and matter to both have "attractive" properties under gravity. General relativity describes gravity in terms of a valence 2 tensor $g_{\mu\nu}$. Upon quantization one would therefore expect a spin $2$ particle. The propagator might look something like
$$(g_{\mu\rho}g_{\nu\sigma}+g_{\mu\sigma}g_{\nu\rho})\frac{i}{q^2+i\epsilon}$$
which in the nonrelativistic limit yields a universally attractive potential by comparing low energy scattering with the QM Born approximation. For more details see Peskin and Schroeder page 126.