(Caveat: I am not a financier).
1) Real answer? No-one knows how prices are adjusted. If anyone figured that out they'd briefly become rich until everyone else caught on and the advantage of knowing is lost into irrelevancy.
2) I take it you're not going to go into differing bid/ask prices (what the buyer will pay versus what the seller will receive); the difference is paid to the broker (the exchange has to make their money somehow). The difference in prices comes from buyers making bid offers and sellers making ask offers, and the two parties having different ideas what they're worth. The broker then matches sufficiently high bids that intersect with sufficiently low asks, takes the bid/ask prices that held when the respective offers were made, charges the buyer the bid price, pays the seller the ask price, and pockets the difference. So you'll need to timestamp offers and price history, and maybe add lifetime instructions to the offers so that they expire if not matched soon enough.
Um, something something buy/sell spread something liquidity something.
To keep things simple: 3) the other will see the increased price (whatever that turns out to be). But remember that they're only making an offer: they get to say what price they're willing to pay when they make that offer. Remember that buyers can only purchase shares that are being sold: sellers choose what price tag to put on their shares (maybe a range of prices within which they're willing to deal), and buyers make offers (ditto). If your buy orders are too low or sell orders are too high, you won't find any takers.
But stocks often aren't traded so bluntly. You can have "put" or "call" options, where you buy the option to buy a certain number of shares at a certain price at a certain time in the future (the price could be the one when the option is bought, or the price on maturity, it could have clauses to automatically revoke or go through with the share purchase if and when various conditions are met). And of course options can be bought and sold. Sometimes optionally. Often bundled, with the whole package being bought or sold based on the performance of its constituents. Entire programming languages have been designed just to describe the various instruments in play—and that's a scary thought.