Zeta - Nobody can really answer that question but you. The first question I would ask you is this - was it a felony DUI? If not, then obviously you don't have to put that down on your application. If it's yes, then it's your call. Some places will run a check, and others won't. You say it's dropped off your record, but I'm sure it will still show up in a lot of searches.
As for the other issue, Deety pretty much explained it. Standard insurance companies, such as State Farm, Allstate, Grange, American Family, Motorists, Erie, etc., will go back 5 years when deciding whether or not to give you a new policy. The non-standard companies, such as Geico, American General, Safe Auto, etc., will only rate you for the last three years, but they will also check back 5-10. And of course you'll be paying more.
To answer the original question, no, you can't do anything about it. It is a part of your driving and criminal record, and those records are almost always public. Even if they weren't, your employer can obtain them with a signature from you, or simply not hire you if you don't give them permission.