How many fields that we know of permeate the universe?

In quantum field theory it is actually not obvious how many fields there are since fields can have components. If we have two fields $A$ and $B$, we can consider them to be merely components of the same field. Or reversely, if $A_1$ and $A_2$ are components of a field we can relabel them $A$ and $B$.

However, it rubs physicists the wrong way to split fields in ways that go against special relativity. Special relativity dictates that for observers with coordinate axes rotated relative to each other, the spin components mix. (If you turn upside down, spin up becomes spin down!) So "the spin up positron" is not a specification of a field component that physicists like. Spin up according to whom? Special relativity also, in a sense, mixes positrons and electrons, so all four possibilities -- spin up/down positron, spin up/down electron -- are considered merely components of the same field, and which component is which depends on who you ask.

Splitting quarks and gluons by color is likewise unphysical. The color of a quark is not even accessible to experiment.

Thus in the Standard Model we have, breaking down into the smallest components allowed by relativity (and gauge invariance),

• 3 lepton fields (electron, muon, taon)
• 3 neutrino fields
• 1 Higgs scalar field
• 3 weak gauge boson fields: the $W^+$, $W^-$ and $Z$
• 1 electromagnetic field
• 1 gluon field
• 6 quark fields

In nature there may be more fields that we have not yet discovered. For example, whatever dark matter is, it is not any of the above. There may exist quarks even heavier than the top that we have not yet seen in accelerators. A lot of people have thought about models with more than one Higgs field. If supersymmetry is realized in nature, we have to double the list. In principle, we should also add a gravitational field to the list, but quantum field theory almost certainly isn't the correct tool for gravitation, so I hesitate.

Every particle has a corresponding field that permeates all of space in the same way the Higgs has a field that does so.

The spin up electron. The spin down electron. The spin up positron. The spin down positron.

The up quarks (all three colors and both spins). The down quark (all three colors and both spins). Same for the charm, strange, top and bottom. And double that because all those quarks each have an antiparticle with the corresponding anticolor and opposite electric charge just like the electron had its antiparticle, the positron.

Then there two more leptons like the electron the muon and the tau lepton (each has two spins and an antiparticle with opposite electric charge).

That's all the fermions that have electric charge. Then there are the eight gluons and they would have three spins each but since they are massless they have two helicity states instead, and they are their own antiparticles)

The gluons are also bosons like the photon, there are only two photon fields, one for each helicity (there would be three spins but the photon is also massless)

There are yet more bosons, the $W^+,$ $W^-,$ and $Z$ each of them have three spins. And the neutrinos are the chargeless fermions and the chargeless leptons. There is one for each of the charged leptons (one for the electron, one for the positron, one for the tao and one for its antiparticle, one for the muon and one for its antiparticle).

Those are the ones we've seen, some people like to predict more. That's already quite a lot and the Higgs is not special (well it is the only spin 0 we've seen so we didn't have to have multiple versions for spin or helicity).

If there is a graviton that would be another.

If someone is talking about early days then I think the idea is the Higgs moves into a lower energy state very early on.

What makes the case of the Higgs field different from that of other particles is that the Higgs field in the vacuum has a nonzero expectation value. So, if the electromagnetic field is in its lowest energy state then that means that the field strength will be zero on average (there are still quantum fluctuations, but on average it is zero). But for the Higgs field this is different, the lowest energy state of the Higgs field is obtained not for zero field strength but for a finite field strength.

In inflation theory one postulates the existence of the Inflaton which would also have a non-zero vacuum expectation value.