Most (all?) bills come with a form to fill out your credit/debt card info and mail back to them. What are the limitations on what they can do with one's card and how long they can hold the info? Like, for example, could they automatically charge the next debt without asking? Or could they just hold the info for a period after using it? Even that bothers me, because it's just another company with possibly questionable security practices that could end up letting my info get stolen. If important, I'm asking about US laws.
In the US, and possibly elsewhere, the main source of requirements for how vendors handle credit card information is the card processors themselves, via the PCI Security Standards Council at https://www.pcisecuritystandards.org/. From what I can see by looking at the standard, and I'm not an expert by any means, the vendor is only required to restrict access to the cardholder data; there's no specific requirement for how long they can keep it.
Of course, this is by no means a guarantee, as we've seen from the Target credit card breach and the like.
So, what can you do about it? Two things come to mind, one responsive and one preventative.
The responsive thing is to monitor your credit card statements, and immediately dispute anything you think is in error. There are a bunch of questions on this site talking about how to do that and what the rules are. You should be doing this anyway.
The preventative thing you can do is use a one-time credit card number in these situations. Some larger banks offer this service, but you'll have to check with your card issuer. They can be called virtual numbers as well.