I have been furloughed from a regional and am applying for jobs. I have 2 moving violations in the last 3 years (speeding and a car-pool lane violation). There are stories to go with each (if I am asked in an interview), but they don't matter; they are on my record. Most of the apps ask about driving record. Is this going to be enough to disqualify me?

On a related question, I went to driving school for one of them, but it's still on my record. If I get it off, do I still need to report it? Is it really invisible or just noted as driving school dismissal?

Not typically - they're more interested in whether or not breaking the law is a trend. If there are other areas in the interview where you are weak then this may compound the issue, resulting in a no-go.
If there is a paper trail then absolutely yes, you must disclose it. If there is no paper trail then it becomes an ethical question that only you can answer. To answer whether or not it is visible - if you can see it they can see it. If it is an egregious offense then it may be reported to the NDR and they will see it when they request those records via your release form. But the only way for them to get their hands on your state driving record is to ask you to bring it with you, mail it to them or fax it to them. There is no release form for state records.

Thanks for the quick response. I feel better now.