Question: If the Higgs field gives mass to particles, and the Higgs boson itself has mass, does this mean there is some kind of self-interaction?

Also, does the Higgs Boson have zero rest mass and so move at light-speed?

Answer: 
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Yes, the Higgs field is self-interacting and, to the extent I understand it, it is this self-interaction and particularly, its form, that allows the Higgs field to "condense" by giving the lowest energy states of the field a non-zero expectation value. But the Higgs fields have electroweak charge.

So, essentially, space, when the Higgs field is in a ground state, is an electroweak superconductor that "breaks" ("hides", "screens") the electroweak symmetry down to the observed electromagnetic symmetry.

In addition to the electroweak gauge field interactions with the Higgs field, the fermion (matter) fields also couple to the Higgs fields via a so-called [Yukawa interaction](http://en.wikipedia.org/wiki/Yukawa_interaction) which is, in The Standard Model, the origin of matter's mass.
The Higgs field can be thought of as a giant field of energy, with the Higgs Bosons being excitations in that field.

This field interacts with particles to give them mass, and this is similar to how the Higgs Boson is 'created'. When a particle causes lots of fluctuations in the Higgs field, it can be thought to be 'interacting' with it. This is not unlike the fluctuations in the field responsible for the existence of the Higgs Boson.

So yes, the Higgs Field can be considered 'self interacting'