Its antiparticle is itself, like photon is anti-photon.
Antiparticle isn't a de-excitation of a field (well... maybe a little in the Dirac sea interpretation, but it's a not a good interpretation at all...).
Yes, the Higgs boson self-interacts with itself. It can interact splitting a boson into two (or merging two into one), splitting in three (or colliding with other, or merging three into one), and the combinations that arise from that (because quantum fields "do what they want", "have free will", "are like fairies but with well defined tastes").
For really understanding this, read peskin-schroeder "an introduction to Quantum Field Theory".