Depending on the depth of the background search and DMV search the DUI would appear on a full motor vehicle abstract search since those events do not disappear. Good luck, and perhaps you should inform them before it appears that you were trying to hide something.
With regard to CT finding out about your NY conviction and what action it will take as a NY attorney I can not be sure.
The NY law is that you must install an ignition interlock device on any car you own or operate whether you live in NY or not. However there is an employer exception. See below.
NY VTL 1198 (8) states that If a person is required to operate a motor vehicle owned by said person's employer in the course and scope of his or her employment, the person may operate...
The court should appoint you a public defender to represent you if you can not afford an attorney. If you choose after speaking with your attorney you have the right to challenge the case and go to trial. If you were to be found not guilty then the case will be sealed as if it never happened. Another alternative is to plead not guilty and discuss with your attorney your rights as it they pertain to a disposition with regard to youthful offender status. Another alternative may be an ACD....
Witnesses do in some instances make statements under oath, under penalty of perjury or in making false statements punishable as a crime. However, in my experience it is rare that a witness would be charged by the prosecutor or police with perjury or making a false statement in a domestic violence context.
While some cases can be resolved quickly, you do not always get the best result by doing so. The attorney you have representing you can contact the prosecutor to try to work out a resolution for the next date. There are many issues and facts that go into resolving cases so please communicate with your attorney to get your priorities conveyed.