"I am living in the U.S., and I have been a U.S. citizen for the past four months. I won the green card through the lottery seven years ago and I am very happy here.  I have a brother back home in Ireland who would like to come here and work for an indefinite period of time. He is 28 years old and lived in London for two years. During that time he was arrested because he had a fight. He was convicted but did not serve jail time.  He was also arrested in Ireland on a drunk driving charge, but he didn’t hurt anyone and didn’t serve jail time.  I would like to sponsor him to come out here, but I’m aware that his background could stand against him. What should we do? Could I sponsor him? Can he come out to the U.S. and stay while we do the sponsorship.  He has stopped drinking and has really straightened out his life and he would do well here.”

Before getting into his convictions, you don’t mention if you’re aware of the lengthy amount of time it would take for you to sponsor your brother for a green card.

U.S. citizens can sponsor siblings for permanent residence, but the fourth category family preference category that allows them to do so is heavily oversubscribed.  The processing time is at least 10 years, so it’s an option that should be used for long-term planning purposes.

As regards to your brother’s criminal record, it’s quite likely that he’d have a problem overcoming his two convictions.  When criminal convictions are involved in an immigration case consulting with a skilled U.S. immigration attorney is a must, so bear that in mind if you eventually choose to file a fourth preference petition for your brother.

Before briefly discussing how criminal convictions affect admissions to the U.S., the answer to your last question is no, your brother would not be able to stay in the U.S. while the sponsorship process was going on.  

Because of the length of time involved he would lapse into undocumented status, which would then trigger a 10-year bar to returning here after he departed for green card processing back in Dublin. 
But first and foremost, he could be barred from entering the U.S. even as a visitor because of the two convictions.

He would be unable to enter the country on the visa waiver program because of the convictions, and would have to apply for a B-2 tourist visa.  (Anyone who has been arrested is ineligible to use the visa waiver program.)  

Your brother would have to present all the court records for both cases, and would have to be interviewed by an officer at the U.S. Embassy.

At the B-2 visa interview, the embassy might choose to refer your brother to an approved physician for further evaluation to see if alcohol abuse is present.

Criminal convictions are taken very seriously when it comes to visa issuance – some of them can result in permanent bars to entering the U.S.  A sole drunk driving conviction that didn’t result in injury could likely be overcome; likewise a simple assault.  Two convictions present a more complex situation. 

Again, before you and your brother do anything, you should definitely consult with an attorney.