Do all the particles acquire mass in the Standard Model due to the Higgs mechanism only? I know that a mass term for an intermediate boson is not compatible with the gauge symmetry. But in principle a mass term for the electron field does not violate a gauge symmetry. However to build an electroweak theory consistent with the observation of the non-conservation of the parity of the neutrino, the electron mass term could not be included and it also acquire mass due to the Higgs mechanism.
Are there some Standard Model particles having an explicit mass term, or all acquire mass as a result of spontaneous breaking of the gauge symmetry and its coupling with the Higgs field?
 A: First, a mass term for electron (or any fermion) does violate the gauge symmetry because, chiral right handed components (singlet) transform differently than chiral left handed components (doublet) under SU(2). So the Lagrangian cannot be invariant under the gauge transformation (transformation of doublet cannot compensate the transformation of singlet).
Second, in the standard model, all fermions acquire their mass thanks to the Higgs mechanism (via Yukawa couplings). The case of the neutrinos is still an open question but in principle, if they are Dirac fermions (as all other fermions), they would be treated the same way.  
A: All the experimentally observed particles that are believed to be fundamental particles, but for neutrinos, in the Standard Model get a mass from the interaction with the Higgs boson. Please note that this aspect of the dynamics of the Standard Model is still under experimental investigation, despite there is data that clearly points towards this direction. 
For neutrinos the situation has not yet been cleared by experiments.
A: In any theory with chiral fermions where Left and Right fermions don't have the same charges such fermions cannot have an explicit mass therm and can acquire mass only with spontaneous symmetry breaking. In the standard Model all fermions except Right handed neutrinos are charged under the SM gauge group and all L/R counterparts live in different representations of the gauge group, so gauge invariance forbids any explicit mass term, except for a Majorana mass term for the right-handed neutrinos, that still violates lepton number
