In a circuit, does a battery create electrons which it 'injects' into the copper wire? Are electrons created from a battery and then 'injected' into the copper wire?
If no, does that mean the battery loses power because the copper wire has run out of free electrons?
 A: 
Are electrons created from a battery and then 'injected' into the copper wire? If no, then does that mean that the battery looses power because the copper wire has run out of free electrons?

Sort of - while you can't create an electron, you can free one by a chemical process, which is exactly what happens in the oxidation and reduction processes at the battery terminals. Similarly at the other terminal, the chemical process will capture an electron and associate it with an ion or molecule.
Elements in metallic form, like copper metal, have free electrons and cannot be made to lose all of them without applying an extraordinary amount of energy. For every electron removed at one end of a wire, one is added at the other (approximately! the wire is also a capacitor, but that is usually excluded from the model.)
A battery that has run flat has run out of suitable chemicals for the redox reactions to take place that free/capture electrons at the terminals. The chemicals in the battery - electrolyte and electrodes - have been chemically transformed.
A: A battery is a combination of cells. Cells work by converting chemical energy, commonly known as Gibbs Free Energy, to electrical energy. Inside a cell, a redox reaction takes place that maintains a voltage across the electrodes.
However, as the reaction in the cell proceeds, the cell-voltage drops and finally becomes zero when the contents of the cell achieve equillibrium. That is when you say the battery is dead.
Wikipedia has an article describing the changes. Also, you may read this to get an elementary idea of how a cell works.
Also, your argument of copper wire running out of free electrons is completely invalid otherwise the wire would attain a net positive charge. The electrons are always in motion. Only the potential difference changes which changes the value of current. By Ohm's law, you can justify this.
On the microscopic level, due to decrease in potential energy, the electric field inside the wire decreases. So,the drift speed of electrons decrease which result in a decrease of current. You may study more about it here and here
A: No, electrons are not created. Instead they are moving down a potential difference.
When the battery loses power, it's because there's no longer a potential difference between its terminals, not because it's run out of electrons.
A: When there is no current or battery in  the conductor atoms are neutral, all the electrons are around the nuclei of the atoms and total charge is zero.
This may help:
 when there is current in a circuit , it means electrons have been  dragged out of their orbitals by the applied voltage  and have a drift velocity towards the attracting voltage.

In a charged battery the chemical process has concentrated a lot of electrons from its medium to the negative pole , which depletes the system making the other pole positive. When the battery closes a circuit the  electrons start drifting , as seen in the first link.
The electrons move a small distances,according to the induced drift velocity, similar to a stadium wave , where there is motion in one direction, but nobody moves far in that direction.
