How does a battery ( primary cell ) work?
is an electric field already there, before connecting it to a wire?
Yes otherwise no current would be able to flow through a load connected to the battery contacts. The electric field is what makes the electrons move and the moving of the electrons is what contains the energy.
Electrons have a negative charge, so an excess of electrons is present on the negative or Cathode (-) connection of the battery. On the positive or Anode (+) terminal, there is an equal but positive charge (due to missing electrons) present.
Note that the total charge is zero, so the amount of surplus electrons at the (-) contact is equal to the amount missing at the (+) contact.
This charge (electrons or those missing electrons) is actually insignificant, it does not contain much energy. The significant part is that if you connect the (+) and (-) contacts, a current can flow because the charges want to neutralize.
However, the chemistry going around in the battery then starts "moving" **ions* (which can have one or more surplus or missing electrons) from the (+) contact to the (-) contact. This process consumes certain chemical substances and forming others while electrons are "made free" and end up at the (-) contact (the Cathode).
The electrons do flow inside the battery; they are transported by the ions from anode (+) to cathode (-), completing the current loop. To make them flow this way, energy is needed. This energy comes from the Chemical reaction going on in the battery.
If there is no load connected to the battery, the electrons remain stuck at the Cathode. They accumulate there until a certain voltage is reached, when the Chemical reaction (which produces the electrons) slows down and stops. The chemical reaction stops because it cannot "deliver" the "free" electrons anymore. It can't deliver the free electrons anymore because it requires increasingly more energy for as the potential increases. Once a given potential is reached, the process stops. When a load is connected, it lowers the potential (slightly), which allows more electrons to be "freed", which starts the process again.
The rate of electrons generated depends on the current through the load. Look up the definitions of electric current, its relation to charge, and the charge of a single electron. Then you can calculate how many electrons are involved in making a current of 1 Ampere flow for example (1 Ampere corresponds to 6.25 × 1018 electrons per second).
Batteries should not burn out; if they burn, you're using them wrong. They are used up or depleted. That's because the Chemical reaction needed uses up the Chemicals inside then battery and replaces them with something else—like any Chemical reaction.
In Primary batteries, there is no way back; when the needed chemicals are used up, the battery has become useless.
In rechargeable batteries, the Chemical process can be reversed by forcing the electrons (on the back of the Ions) to flow the opposite way (from Cathode to Anode) inside the battery.
How long a battery lasts depends on how you use it. A situation that quickly drains the battery uses less total energy than one where the load needs a small current. Look in battery datasheets to see the difference; there can be up to a factor 5 difference in available energy!
Note that I am not a battery expert, if there are any flaws in my reasoning please mention them in a comment.
Normally I would say "ignore the electrons" while trying to understand electronics, but this is one of those cases where they actually matter.
I'll use a standard alkaline battery as my example. Chemistry:
Zn(s) + 2OH−(aq) → ZnO(s) + H2O(l) + 2e− [Eoxidation° = +1.28 V]
2MnO2(s) + H2O(l) + 2e− → Mn2O3(s) + 2OH−(aq) [Ereduction° = +0.15 V]
There are three elements. An anode (zinc), connected to the battery negative terminal; a cathode (manganese oxide) connected to battery positive, and an electrolyte (potassium hydroxide, which is the "alkali" of the name). The electrolyte in any battery is a liquid which sits between the electrodes.
"Electronegativity" is the tendency of chemicals to attract or lose electrons. The top reaction is quite keen to shed its two electrons and dump them onto the anode. It does this until a field of approximately 1.28V builds up between the electrode and the solution, at which point the reaction no longer has enough energy to overcome the field and push electrons onto the anode. Similarly the other end builds up a small field.
Note that the electrolyte is not conductive, at least not in the same way that metals are. There aren't electrons floating freely in the electrolyte. There are however ions, carrying a positive (potassium) or negative (hydroxide) charge. The ions are free to travel from one electrode to the other.
When does it run out? When the Zinc has all been oxidised to zinc oxide, or manganese dioxide has run out.