At first that seemed to answer the question but thinking on about it instead it brings more things into question, like how?
Yes, it does.
People always say "supply and demand", but this doesn't explain the mechanics.
When you look at "level 2" it becomes apparent: that shows what's for sale, at what prices. To give a very simplified example, if you want to buy 1,000 shares in a company, there might be 400 for sale at $28.00 and then 400 more at $28.05 and you have to take the last 200 of your order at $28.10, so the act of your buying (when the price was $28.00, or $27.98 - $28.02) has left the market with the next ones for sale now at $28.10, or perhaps $28.08 - $28.12 if you want the bid/ask shown.
I don't trade stocks myself, and my numbers might be inappropriate, but I think that example demonstrates the mechanics of why/how "demand" (better known as "buying pressure") actually increases the price at which they're available, for the next buyer after you? Does that make sense?
Is there one super trading computer that is programmed to raise or lower the price of all stocks according to supply and demand?
Not quite, and it's not as simple as that. My example above illustrates a simplified version of a stock for sale at only one central exchange. But if they're for sale in more than one place, there are also automated trading algorithms seeking arbitrage opportunities and so on, which normalize/balance different supplies anyway.
And whatever does it, is it done proportionally the same for all stocks?
No, not at all.
Will all stocks that are currently $1 go up the same amount if 10K shares are bought, and drop the same amount if 10K shares are sold, etc?
No, not at all. That depends on liquidity. Buying 1,000 shares in Microsoft isn't going to move the price at all, because there'll be a huge number of them for sale at the first price you can take, whereas if you want to buy 1,000 shares in a tiny mining company in some small country, that company has far fewer available in the first place, and they're not traded much, so even your 1,000 share deal will effectively raise the price a little (I'm simplifying, but you can appreciate the principle, from what I'm saying?).
And those decisions are communicated to all trading establishments (or whatever they're called) and related establishments all over the planet, simultaneously, during all of every business day?
There's an element of that, in the computer/internet age, yes.
Tell me whether what I've said above makes sense to you, David, and if not I'll try to explain it a different way.