One of the eligibility requirements to obtain naturalized U.S. citizenship is that the applicant be a person of "good moral character," particularly in the last five years before applying. With a DUI (or DWI, OWI, OUI, DUID, etc.) on your record, it is indeed possible for the officer of U.S. Citizenship and Immigration Services (USCIS) to deny your application on the basis that you lack the required good moral character. However, the decision is not automatic.
No cut-and-dried definition of “good moral character” can be found in the U.S. immigration laws. Most courts describe it as character measuring up to the standards of average citizens of the community where you live. That doesn’t mean that the applicant has to have done anything special. For most applicants, it’s enough to prove that they haven’t broken the law, and are productive members of their community.
Of course, with a DUI on record, you’ve got a strike against you. The first thing USCIS will likely want to know about are the facts of the DUI. Was it your first one? How impaired were you while driving? Were there any aggravating circumstances, such as children in the car, or injuries to others? Every person’s fact situation is different, which is part of why we say there’s no automatic decision on naturalization applicants with DUIs.
No matter the facts of your DUI, you will also want to try to outweigh this incident with evidence that, in every other aspect or period of your life, you’ve behaved well. For example, including evidence of involvement in your community or church/religious organization, your record as a good employee, and your efforts on behalf of your family and children, and so forth, will be helpful.
YES. Under no circumstances should anyone knowingly omit any such information on their application for naturalization. The N-400 form specifically asks about any criminal convictions or charges, and if the applicant is found to have purposely omitted a DUI conviction or charge, then the consequences will quickly become much more severe. In any case, the DUI information will turn up during the background check.
It's not likely, but applying for naturalized citizenship with a criminal record could potentially jeopardize your permanent resident status.
This is not a situation where you should just submit your N-400 application and hope for the best. Get an attorney to analyze your situation and help you decide whether to apply for citizenship now (as opposed to waiting until the DUI is farther in your past). If you decide to apply now, the attorney can also help you present your good moral character in the most positive way possible, and attend your naturalization interview with you.