I'd like to be charged every 16th of the month but i'm charged every 10th. Is there an option I can change in the backend of Stadia or do I have to call customer support, or even cancel the service and sign up again on the desired billing date,
no this is not possible. You can't change the payment/billing date for any Google subscription.
What you could try would be cancel Stadia Pro and when it expires, try to wait to the date you would like to be charged and then subscribe again, but this would definetely interrupt your Stadia Pro experience, because you might be some days without Stadia Pro. I just mention it, because it might be an option you could try.
What you could try as well is if you can pay early (according to this article), but keep in mind this might not work for every subscription and therefore might not work for Stadia Pro. But if the option exists, you could try this.
even when you do it on the same day, there might be a short period in which you basically don't have Stadia Pro anymore, which does only mean that you will be unable to access your previously claimed games. You can still play games though that you have purchased.
Also your data of those games remains intact, even after Stadia Pro runs out, so you don't lose anything. You just cannot access it until you either renew Stadia Pro or purchase the game.
What happens to my save data if I purchase a game I previously claimed?
If you cancel your Stadia Pro subscription and it expires, your save data for games that you previously claimed is not deleted. If you purchase a game that you had previously claimed, your save data will remain intact. (Source: This article)