I'm not sure about the interest if you aren't using margin, I wouldn't think there would be any, but your brokerage should be clear about the rate somewhere on their site.
Unless you cancel your order, or it hits its expiry, it will remain open even with your computer or trading platform shut down. I'll attach an example of available times in force on DAS trader. For example, if you select DAY as the expiry, unless that order gets filled, it will cancel at 4pm Eastern. If you select GTC, the order will remain open until it's filled or you manually cancel it.