Any process (which results to some particles) that doesn't violate your assumed conservation laws is possible... The probability is taken by theoretical considerations.
For example in the collision of two protons, 2 gluons may "fuse" to produce a Higgs particle which will later on decay to let's say 2 photons, 4 leptons (by decaying to ZZ) , 2 leptons (prefering heaviest leptons, eg ditau), or even to quarks (especially heavy b-quarks, which will hadronize) etc...
Of course the production of the HIggs is way less probable than the gluons fusing to produce other gluons or quarks... That's 1 reason why it was so amazing that the Higgs was found: within such a large background, like finding a needle in the haystack (if not less probable).I don't understand the question.
If there are very massive particles, probing higher energies can allow you to search for them...
If for example you collide electron/positrons at 1GeV, you will probably not see Z being produced (needing 90GeV just to be produced at rest)...only if you are able to see its tails around the pole, but still at 3GeV I don't think you can.
Nobody assumes that Higgs bosons exist everywhere... the Higgs field does, as all the other fields, but fields are nothing but mathematical constructs which allow you to describe particle physics...