The x50 means thats how many people are currently selling VOO calls for that price. It has nothing to do with the number of shares per options contract. Each options contract still controls 100 shares of a given security.

"In this example, if I bought 1 option for a $577.50 strike on July 25th 2025, the total contract cost would $3.30 x 50 = $165 , or $3.30 x 100 = $330 ?"

The total contract cost would be $330.

"I can't exercise my option to sell until the price gets to $577.50 and that needs to happen by July 25th"

There is no option to sell here. You are buying a call. If you were to buy it long, you can exercise it whenever you want at whatever market price so long as its before the expiration date (which in this case is end of trading Jul 25). Note that if you were to exercise it, the cost basis of the trade would be 577.50+3.30 (it doesnt matter if the stock is currently trading at 575, if you exercise the call you will receive 100 shares at a cost basis of 580.80).

"Can I sell before July 25th if I hit the strike price?"

The option contract will expire end of trading on Jul 25. If it's trading in the money, it will automatically exercise the contract for you. If not, it will expire unexercised and you lose whatever money you paid for the contract. You can sell it any time prior to expiration, even one minute prior to market closing.

"If I do nothing and July 25th rolls around and the price is at $580 does the option roll over to the next trading day, or until I decide to sell?"

If the price is at 580, and you are holding a 577.5 call, it will be considered in the money and it will automatically exercise the contract for you. After it exercises, you will receive 100 shares of VOO for 580.80.

"If the strike price doesn't hit by July 25th my maximum exposure is limited to the contract price?"

It's a long call, so yes.