The answer hinges on the difference between quantum mechanics and quantum field theory, the latter of which is used to describe the physics of elementary particles. In quantum mechanics you sum over all the paths a particle might take to calculate the probability that a particle might be in a given place; there is no concept of "the path the particle took". In quantum field theory, you sum over all the paths a particle might take *and* all possible interactions it can have; there is no concept of "only interacted twice".

For the case of virtual particle momentum you talked about it gets a little more

complicated. The formalism requires you to sum (actually integrate) over all possible momentum, however high. Some such integrals will be ill defined, however (give infinite answers). In such cases, the sum/integral effectively restricted to a finite range of momenta, leaving out momenta over a certain value. The particular maximum

is arbitrary, as long as it is much higher than any momentum you are dealing with, with the strength of the interactions (ans values of the masses, etc..) being adjusted, as this maximum momentum is adjusted to keep all the answers the same.

A second point is that the idea of "interacting with the Higgs giving mass" doesn't really tally with your picture of interacting particles. The picture you present is

what I would call "perturbative": particles move along and every so often they interact, with some fixed probability per interaction such that having "many" interactions is unlikely. From the quantum field theory point of view you can try and apply perturbation theory for excitations about the vacuum (lowest energy particle/field configuration). It doesn't always converge (i.e. the case of interacting 10 times may be way more important than the case of 1; 100 more than 10 etc...), but it might be a good picture. For the standard model, it is a good picture for electromagnetism and the weak force, but not for the strong force (QCD) at low momenta. Adding the Yukawa couplings to the higgs (the way the W and Z are given mass), however, is a dramatic enough change to the physics that it cannot be described by perturbation theory, and instead should be thought of as a change in the vacuum; if you want to have a mental picture of a W moving along and rarely interacting with a higgs, it should be a W that already has mass.