How does current flow work?
Does the electrons at the anode (negative end) of the battery flow through the copper to the cathode end or is it only the copper atoms electrons that flow to the cathode end of the battery or does both happen?

Not at all. Electrons flow from the negative end of the battery to the positive end. The current convention says that current flows from plus to minus, that's just a convention though, in fatct electric current flows in the opposite way electron current flow. And no, not only the "copper electrons" contribute to current, that's because electrons are actually indistinguishable: one enters from the cathode, another one exits the anode: not the same electron, but they "belong" to the same current.

Does the electrons at the anode end of the battery ever strike the copper atoms valence electrons?

Yes, that probably happen but is very, very unlikely. Classical electron radius is some 2.8fm while copper atom radius is 128pm. That means that hitting an electron with another electron is like feeding a thread through a needle's eye, and we are randomly trying on a surface that's 2000 square meters.

And if so is that what causes heat? Or is heat produced when the copper atoms strike each other when they randomly zigzag to the positive end of the terminal?

Well it might cause a tiny amount of heat. Heat in a battery is caused by:


chemical reactions happening inside
electric power dissipated in resistive components


The second is due to electrons hitting nuclei of copper or whatever conductor is there. And pay attention, nuclei do not move at all, they just vibrate a bit.

When the copper atoms lose electrons because it's attracted to the positively charged terminal. That copper atom becomes a cation how does current flow if that atom has a net positive charge?

I do not see the problem, current could flow anyway, keep in mind that when an electron "is lost" another comes in, conduction is a sort of relay race. See this other answer (I know auto quoting is bad but apparently the community liked that one).