I have a silly question - what would happen if weak bosons wouldn't interact with higgs field? What would happen to weak interactions? How it would affect mass of other particles?
what would happen if weak bosons wouldn't interact with higgs field
The Higgs field is a pair of complex scalar fields that transform as an electroweak doublet.
So, on this view, if the fields don't have weak isopin charge and weak hypercharge, they're not components of the Higgs field; they're something else entirely.
Essentially, you're asking what would happen if the Higgs field was not a Higgs field.
Well if you turned the coupling off, the Higgs mechanism wouldn't apply, and the symmetry would never be spontaneously broken. And then rather than having massive fields, you'd have massless fields and the interaction in the Lagrangian that would have been a mass term would instead be a Yukawa interaction. It'd be very similar to what happens at high energies (above the symmetry breaking scale).