The Higgs field is a scalar field and it happens that the vacuum expectation value of that field is non-zero in our universe. It is this non-zero Higgs vacuum expectation value that gives the elementary fermions of the standard model of particle physics their rest mass. Now this Higgs field is a scalar so it is as if there is a single numerical value that specifies this field strength everywhere in space.
If you think of this scalar value as being like a the depth of the water in a swimming pool, then the Higgs Boson is like waves on that water. So a Higgs Boson causes small up and down variations in the Higgs value as it travels through space. And as a particle, it takes 125 GeV of energy to create the Higgs Boson. So the Higgs Boson can only be created where that much energy is available. Cosmic rays hitting the earth have energies that high and much higher so it is certainly possible for cosmic rays to be creating Higgs Boson particles independently of the LHC or any other human particle accelerator. However, these Higgs Bosons are certainly not bombarding every particle all the time.
However, I think what Dr Cox is talking about is the Higgs scalar vacuum expectation value that fills all of space. That is what is giving elementary fermions their rest mass, but this is not at all the same as the Higgs Bosons that were created and detected at the LHC. So I think Dr. Cox was taking some liberties to try to explain this complicated physics to non-physicist audiences.
Conservation of mass says that matter cannot be created or destroyed.
From this answer (below) by Luboš Motl Conservation of Energy, and also, for one example, from the idea of dark energy producing an expansion of the universe, it would seem that conservation of mass/ energy does not hold, at least in the way we view it in classical mechanics. Obviously matter was created at some stage after the Big Event, or we would not be here now.
For different mixtures of matter obeying different equations of state (roughly speaking, with different ratios of pressure and energy density), one will see the total energy increase or decrease or be constant. Generally, the total energy of the Universe will tend to increase as the Universe expands if the Universe is filled with matter of increasingly negative pressure; the total energy will decrease if it is filled with matter of increasingly positive pressure.
Your next point:
If there was no particles with mass in the universe before a Higgs Boson Field swept through the universe then what was there.
If we take the Big Bang Beginnings article as a refence for the evolution of the universe, just after the big bang, initially there may have been purely radiation, which then as the temperature dropped, became a mix of radiation and particles. We don't know exactly what happened or existed before 10$^{-12}$ second.
Image source: Timeline of the Universe
Quark Epoch, from 10$^{-12}$ seconds to 10$^{–6}$ seconds:
Quarks, electrons and neutrinos form in large numbers as the universe cools off to below 10 quadrillion degrees, and the four fundamental forces assume their present forms. Quarks and antiquarks annihilate each other upon contact, but, in a process known as baryogenesis, a surplus of quarks (about one for every billion pairs) survives, which will ultimately combine to form matter.
I don't think anybody knows how close in time (or perhaps simultaneously) the creation of the Higgs Field occurred with respect to the appearance of the 4 separate forces.
Hadron Epoch, from 10$^{–6}$ seconds to 1 second:
The temperature of the universe cools to about a trillion degrees, cool enough to allow quarks to combine to form hadrons (like protons and neutrons). Electrons colliding with protons in the extreme conditions of the Hadron Epoch fuse to form neutrons and give off massless neutrinos, which continue to travel freely through space today, at or near to the speed of light. Some neutrons and neutrinos re-combine into new proton-electron pairs.
Lepton Epoch, from 1 second to 3 minutes:
After the majority (but not all) of hadrons and antihadrons annihilate each other at the end of the Hadron Epoch, leptons (such as electrons) and antileptons (such as positrons) dominate the mass of the universe. As electrons and positrons collide and annihilate each other, energyin the form of photons is freed up, and colliding photons in turn create more electron-positronpairs.
Nucleosynthesis, from 3 minutes to 20 minutes:
The temperature of the universe falls to the point (about a billion degrees) where atomic nuclei can begin to form as protons and neutrons combine through nuclear fusion to form the nucleiof the simple elements of hydrogen, helium and lithium. After about 20 minutes, the temperature and density of the universe has fallen to the point where nuclear fusion cannot continue.
Your next question:
Lets just say that the big bang is was actually when a Higgs Boson Field swept through the tiny universe and created particles that have mass. A universe exists somewhere else and they never had a Higgs Boson Field to give their particles mass. Is that universe just a mess of gluons and photons?
Yes, without a symmetry breaking Higgs Field, all possible particles, in addition to the force carriers you mention above, would have been massless.
I apologise, but I feel the the rest of your questions in your post require a separate answer.
Best Answer
Yes, an electron is just some wave, as you say, in the electron field, as it is for any particle. You can also interpret in a broad sense that a field needs to be perturbed at a particular point in spacetime for you to have a non-zero odd of measuring it a that point, although this simple picture is complicated by quantum phenomenas.
The energy of a decaying particle not only can but needs to end up somewhere. This is conservation of energy! The mechanism are not unknown, they are the possible interactions (read that as forces) between fields, though they are not all clearly understood in their dynamics.
The idea of billard balls particles colliding is really not the best to have in mind when considering QFT. The electron, which is really a wave/excitation in a field, travelling in spacetime in presence of the Higgs field does not need to ''collide'', in a classical view, with a Higgs particle to interact. Keep in mind that these field excitations are not exactly localized, much as a wave is not. What happens is that the electron field interacts with the Higgs field and as seen form the dynamic of the electron field it corresponds to it having a mass. The closest analogy that comes to mind, which is pretty bad: don't give it too much intellectual weight, is of a bullet going through water that acquires a different dynamic behavior by interacting with the surrounding media, but that's as far as it goes.
Your question about the difference between a Higgs particle and another one, is like asking what is the difference between sound and light. They are not excitations of the same medium.
I am sadly not aware of any good and simple analogies for the Higgs mechanism. The closest thing, which is not simple but quite close conceptually speaking, are electrons in crystal having a different effective mass because of their interaction with the crystal lattice. Without using effective field theories, you can model electron wavefunctions moving in the crystal lattice using standard quantum mechanics. From there, you study their dispersion relation which is in essence the equation relating energy and momentum. The dispersion relation, in some cases, will take the a functional form of a free wave from which you can infer an effective mass. You can interpret that as saying that the interaction with the lattice modifies the mass of the free electron.