From my understanding (/me not a physicist), it is rare. It's created only in high-energy situations, and decays extremely quickly. In the very beginning, there were a lot of them, but not for long.
I gleaned from the article that the field is everywhere; I think it's correct to say that all fields are everywhere, but according to the article the Higgs field is distinctive: "The Higgs field has a nonzero vacuum expectation value throughout all of spacetime, meaning there is always some value associated with it, even when no Higgs particles are present."
I guess, FSVO "present". If there were no Higgs particles anywhere, would the Higgs field disappear? I have no idea.
There are two equivalent ways to represent the universe:
1) The average value of the Higgs field is zero and there is a HUGE amount of Higgs bosons everywhere, but all the calculations are horribly^1000000 difficult.
2) The average value of the Higgs field is a constant that is not zero and there are very few Higgs bosons here and there, and the calculations are easy [1].
Obviously physicist prefer the second description, in spite both are equivalent.
There are some technical problems if you imagine that there is a constant everywhere in the universe, and has exactly the same value [2]. So the solution is that it has the "constant" has an average value and allow local variations. The local variations are the Higgs bosons, because the field is quantized.
In a universe where the average value of the Higgs boson is not zero, but there are no Higgs bosons, you get the same technical problems that were solved with the idea of Higgs.
[1] It's easy if you have a PhD in physics, a few years of specialization. I can't do them, but I know people that can.
[2] There are other constants anyway, but they are different... I have no better way to explain it :(.
...which is counterfactual, if you mean "anywhere in spacetime", because there were a lot of Higgs particles, at the very beginning; and those particles exist in spacetime. If there are no Higgs particles in spacetime, that's a different Universe. I guess (/me not a physicist).
This is a good summary. Only thing to add is that the Higgs particles are basically oscillations on top of the background value of the Higgs field, just like photons are oscillations of an EM field. In both cases, the field is the more fundamental object, so your question at the end is backwards :)
Those fields are a mathematical tool to describe what is going on in a specific way - for example making the local character of particle interactions obvious - but there is no evidence that there are actually any fields. Quite the opposite, a good reason to doubt that those fields are real is that they have Gauge redundancies, i.e. one physical state has several different mathematical representations.
So basically a certain energy allows to bring forth certain particles into being through laws of physics. How does the universe know the laws, how is it encoded? Sorry if I am vague!
The universe simply is, and “laws” is what we know about what/how it is. For example, the law: “Things consist of atoms.” The universe does not “know” this law, we do.
