At least in this case, GameStop has filed to raise up to $100,000,000 through an ATM offering (selling shares to the public) [0]. So yes, the recent activity does help the company, in so far as GME is a company that can be helped by getting its hands on more cash.

[0] https://www.sec.gov/Archives/edgar/data/1326380/000119312520...

Did that go through? Releasing new overpriced shares while the price is being obviously raising unrelated to the company performance sounds super dodgy.

They planned to sell them before the current rally (the document is dated December 8). They haven't released details of how many they sold, or on what dates.

It's not clear what view the SEC would take if they fast tracked an ATM offering now. Hertz were told to knock it off, but they were literally trading in bankruptcy. There's a pretty good argument to be made that issuing more shares is if anything the responsible thing to do to in response to increased demand, since it would help to correct the anomalous price rally.

Worked fine for Tesla and Apple.

Not directly. But driving up a company's market cap can make its stock more attractive to a potential acquisition target. Or it can simply sell shares of itself at the new, higher price.

Yes it does, the company can do a stock split and sell more shares.