Publicly traded companies are officially listed and traded on one or more stock exchanges. Stock exchanges are separate entities, which process all the bureaucracy around trades. A trade requires a seller and a buyer to agree on a sale price. If you own a share you 'tell' the exchange that you are willing to sell and at what price. If you are a buyer you 'tell' the exchange what price you are willing to pay. If a match is made then a trade is completed and recorded by the exchange which posts how many shares at what price were traded.
The total number of shares a company has is fixed, known by everyone, widely published, and tracked by the exchange. A trade does not affect the total number of shares, it just transfers ownership from one person to another. Companies can issue additional shares to raise more money, which normally reduces the perceived value of all pre-existing shares since existing shareholders would automatically own less of the company if new shares are issued. This is called dilution. It does happen, but companies have to be careful when and why they would do this so they don't piss off existing shareholders. The opposite is also true in that companies can buy back their own shares, which raises the value of all remaining shares as there are fewer shareholders that would then own the company.
Each day the exchange is open, normally Mon thru Fri, except holidays, people can offer to buy or sell shares of companies traded on that exchange. The opening price for the day is the closing price the previous day (not always because of news, afterhours trading etc). All that means is that the exchange publishes every trade and the close is simply the value paid at the end of the day by the last traders. At all points through the day, the price can go up or down depending on who is willing to buy or sell at what price. It is completely dynamic and the opening or closing prices are no more or less important than at any other point in the day. There is no averaging per se; the published value is what the last buyers and sellers agreed upon, but the very next trade could be higher or lower so current value is always dynamic. Looking at the last few trades can give you an idea of price, but that doesn't guarantee anything.
Currency is valued and traded the same way. A countries central bank is like a company only instead of shares it is money. They can print more money which reduces the value of their currency relative to others. And they can buy and sell debt like a company can buy and sell its own shares. In any case, central banks make certain policy decisions etc, and the results of those decisions play out in currency markets like stock markets.
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