The first thing that you need to know is that these items are civil, not criminal. If you don't pay a bill, that simply means that they'd have a civil claim against you. They could sue you in court, but they can't throw you in jail. The prosecutor is not going to come after you for not paying (unless there's some element of a crime that could be proven, such as fraud). Furthermore, if this happened 10 years ago, the statute of limitations for any crime or civil action would have run (meaning that they can't now take you to court).
A credit check is not performed by the USCIS as part of the green card application process. The background check is looking at national security and criminal background concerns. They're looking to see if you're a threat to national security or if you may be a criminal. They look for "crimes involving moral turpitude" (CIMT). These are "bad" crimes such as murder, rape, drug trafficking, etc... Again, unpaid bills isn't even criminal in the first place, much less a CIMT, so you wouldn't have to worry about it.
Like I said above, the statute of limitations would have run anyway, so if they didn't sue, they can't sue. The "Fair Credit Reporting Act" (FCRA) states that creditors can't report debts past 7 years after the last event (when it was incurred without payment or the last time that something was paid towards it). So if it happened 10 years ago and were not sued, etc... then it wouldn't even show up on your credit. But like I said, credit is not a part of the DV lottery background check in the first place. That's not what they're looking for, and they don't care about your credit.
Hope that clears things up a bit. If you have any other questions, please let me know. If not, and you have not yet, please rate my answer AND press the "submit" button, if applicable. Please note that I don't get any credit for the time and effort that I spent on this answer unless and until you rate it positively (3 or more stars). Look for the stars on your screen (★★★★★). Thank you, ***** ***** luck to you!