These are pretty straight-forward questions. I don't think this is going to take a lot of citation (though I'll try to find pictures of coins for examples), but as a primer to ancient numismatics I want to suggest Howgego's Ancient History from Coins, which was on the American Numismatic Society's Summer Seminar reading list a few years ago, if it's not now. I'm also only going to talk about western coins. Coins in China arose independently and I don't have the background to discuss them.

It's not clear how coinage arose. Herodotus says the Lydians were the first to strike coins. They used electrum (a sliver/gold alloy), and at first the common characteristic was uniform weight, not shape or iconography. The earliest just had a rough image on the reverse (heads) and an incuse mark (a square punch) on the obverse (tails). I pulled some examples from ANS' catalog here. The British Museum has an introduction including several examples of more developed Lydian coins.

We don't know why these coins were struck or who struck them. We have some names but we can't identify them beyond these coins. We don't know if they were individuals or government officials. The fact that they were struck to a weight standard suggests some kind of importance on the amount of metal in them, which might argue for convenience of exchange - if a coin with a certain mark has a known and trusted value, you don't have to measure each one to find out how much it is worth.

We get in the 6th century gold and silver coins identified with Croesus, king of Lydia, and as coinage moved into the Greek world it became an activity of states, and not individuals.

These coin-striking states, as you guessed, did use the metal in the state treasury to strike the coins which they used to pay their armies and bureaucrats, so that when Rome left Britain in 407 CE, the coinage supply completely dried up for a while. Coins penetrated the rest of the economy as soldiers and other state employees used the coins to buy stuff.

And again you guessed right about debasement. If a state had more expenses than they had metal to spend, one solution was to debase the coinage. This would not work for long; merchants would catch on to the inflation and adjust prices accordingly, but if the coinage had a well-established reputation for quality it might take a second. So you find debasements happening in times of say military emergency. Occasionally we speculate about an issue of coinage being for other kinds of emergency: I think Crawford speculated that Roman coinage in the Republic issues by decree of the Senate was issued for special monetary crises and not necessarily things like wars. This is not regular though.

I can discuss this question with regards to the political economy of the Mughal empire in the 16th and 17th centuries, which I have studied in some depth, though I'm sure the basic principles apply to other historical empires as well. Currency acts as an economic lubricant that facilitates the movement and centralization of goods and services, allowing the administration and the military to function smoothly. To better understand this process, it is important to emphasize not only the role of the state in the production and disposal of money, but also the significance of state revenue demand in that same money and the effect that this has on the wider private economy. I will try to illustrate this below as a cyclical series of steps in the specific case of the Mughal empire:

1. First, the Mughal mints (located at major cities) produced money from metal, scrap, or from old coinage. The Mughal mints were open to the general public, so that anyone could bring raw material and have it minted. The most common coins in the Mughal empire were the copper dams and the silver rupees, though some gold coins were in circulation as well.

2. The Mughal government used the money to pay the salaries of imperial servicemen of various kinds who were based in cities and towns throughout the empire. At the provincial (subah) level, the Mughal administrator in charge of fiscal management was the diwan, and each provincial diwan reported to the vizier or diwan-i kul at the center, who was the chief financial administrator of the empire.

3. The Mughal servicemen who received salaries from the government used the money to purchase various goods and services in the settlements in which they were based. Most important of all, the Mughal mansabdars or imperial officers had to use the money to procure foodstuffs and other supplies, since they were responsible for maintaining a certain number of soldiers to be available for the Mughal emperor and the imperial governors or subahdars.

4. To meet the demands of the imperial servicemen, a specialized class of private merchants (often denoted by the term mahajan) moved goods from the producers in the countryside to the consumers in the towns and cities, and sold those goods for money. In the opposite direction, money moved out from the towns and cities into the countryside, because merchants purchased goods from rural producers with money.

5. The peasants in the villages of Mughal India had to sell their produce to merchants in exchange for money, because the Mughal revenue administration demanded that taxes be rendered in money and not in kind. Whether taxation was done directly through amils (state revenue officers) or indirectly through zamindars (hereditary local potentates), the revenue that flowed into the treasury of the diwan was expected to be money.

6. And finally, the fiscal administration under the supervision of the diwan would use the money revenue derived from the peasants to meet expenses, which would facilitate the movement of goods and exchange of money, which would ultimately provide the peasants with the money they would need to pay their taxes, which would enable the Mughal government to repeat the above cycle once again.

Thus, we see how government money creation and government revenue demand in that same money acts as a massive "fiscal pump" which draws resources from the countryside into urban centers via private mercantile actors, and thus satisfies the enormous demands of the Mughal administrative and military apparatus in a reasonable and economical manner. The Mughal mansabdars were charged collectively with maintaining over 150,000 professional cavalrymen, constituting the bedrock of peace and stability in the empire, and this massive military establishment could be effectively maintained without any government micromanagement thanks to the mechanism outlined above.