Yes, okay. Masses are not derived from the higgs boson, they are a function of particles interacting with the higgs field. The higgs boson is just one component of the higgs field.
I am still trying to give my self a simple way of understanding it in my head. This is interesting, but dense information for me :)
Difficult to say without entering deeply into the equations. I should not have used the world field... Well now it is too late... ;)
The Standard Model is a quantum field theory. Which means, you have strictly speaking only fields that interact with each other, and the field are the fundamental objects of the theory. The particles can be more seen as arising from the fields. And the Higgs boson arise from the Higgs field.
You may ask what is going on with the three other degrees of freedom of the Higgs field. A massless vector boson like the photon has two degrees of freedom, and a massive one like the W^+, W^- or Z bosons have three. We need to render three bosons massive... Then the count is right.