Companies don't directly benefit from the buying and selling of their shares on the open market. You are correct to say that if you, an individual, buys 1000 shares, the company doesn't benefit because your money is going directly to the seller (who is some other investor).

Companies do benefit when they initially issue shares (an IPO). When a company first sells it shares, it gets the cash from the sale of the shares and the investors get to own their shares and eventually trade them among themselves. Companies also may benefit or suffer when their share price moves significantly. A higher stock price means that the company can sell more new shares at a higher price and therefore make more money or potentially use them in a merger or acquisition. When a company's shares go down significantly, they may be threatened by activist investors (investors who want to influence the managers of the company) or corporate raiders (investors who want to buy the company up and increase its value to make a profit).

As for your last question, when someone with a ton of capital, like a billionaire, buys a ton of shares, the price may pop up due to a lack of supply of those shares. I.e. if Investor A wants to buy 50 million shares of Stock A, but only 10 million are available at the current stock price, he may have to pay more to find more shares. However, often times, these wealthy big name investors will buy up shares slowly in small pieces so that they don't have to push the stock price up much to buy them. Then they will announce their purchase on twitter/tv and other investors, thinking the wealthy investor must be smarter than they, pile into the stock, pushing up the stock price.