The search for Sir Rudolf Bing and his wife Lady Carroll Bing has ended. Police in Scotland said the couple were found yesterday at a hotel in Dunbar, a fishing port east of Edinburgh. The 85-year-old Bing, former head of New York's Metropolitan Opera, and his 47-year-old wife had not been seen by their lawyer in more than three weeks, and on Wednesday the couple had been officially listed as missing.

Bing, who has disappeared several times since moving to Britain earlier this year, reportedly suffers from Alzheimer's disease. A New York judge has ruled Bing incompetent to manage his $900,000 estate and has permitted a court-appointed conservator to start proceedings to annul his nine-month-old marriage.

Out and About Jody Powell says he believes that, had circumstances been different, Jimmy Carter's political career could have been damaged by scandal in much the same way as Gary Hart's. Speaking before the Maine Safety Council in Portland Wednesday, Powell said that two months before the 1976 presidential election, Republicans found a personal letter to Carter from a woman. An attempt was made to interest the press in a story about Carter having an affair, Powell said, even though the letter indicated "only a close but innocent friendship." Fortunately, he added, the woman and her husband were close friends of the Carters and quickly issued a firm denial. But Powell, who said yesterday that he worries about the explosive area the press has wandered into in the Hart case, told his Maine audience, "I wondered then what would have happened if Jimmy and Rosalynn had had a falling-out with those people years before. What if they hadn't been willing to clear it up, or turned into Republicans or some God-awful thing like that" ...

Surgeon General C. Everett Koop, a strong advocate of safe sex, will be in Beverly Hills today to discuss the entertainment industry in the age of AIDS with a group of more than 250 television producers, writers and directors. Moviemakers are being more careful about portraying sex. The most recent James Bond thriller had 007 sharing his affections with only one woman, and the subject of safe sex comes up more frequently in other films. The Center for Population Options, a Washington-based organization, arranged for Koop to meet with the television representatives because it believes the industry has a strong influence on what teen-agers know about sexuality ...

It was big stuff for Janesville, Wis.: Daniel P. Federici, keyboardist in Bruce Springsteen's E Street Band, married Kathy Helmeid Wednesday at Asbury United Methodist Church, and the Boss was there to be best man. Springsteen arrived in a white limousine and didn't acknowledge the crowd chanting "We want Bruce" as guests began leaving the church ...

Actor Lee Marvin, who died of a heart attack Aug. 29 at age 63, will have his wish fulfilled. He will be buried at Arlington National Cemetery. The World War II Marine veteran made 21 landings in battle zones, was wounded on Saipan and spent 13 months in a hospital. He will be buried with military honors Oct. 7 ...

How soon they forget. Shari Theismann, who is now living back home in South Bend, Ind., to be near her mother, was in town yesterday for a fashion show. Asked why there was little more than a passing reference to her in the book written by her husband of 14 years, Joe Theismann, she replied: "Joe must have been involved in immaculate conceptions, because someone in the house had three children." Broken legs seem to run in the family. She said her son Joey is receiving therapy at Notre Dame University as he recovers from a broken leg ...

It doesn't take long for Washington's own brand of humor to develop. No sooner had Sen. Joseph Biden's plagiarism problems surfaced than this story began making the rounds: "Rep. Pat Schroeder calls the presidential hopeful to inform him she is entering the race. Biden responds, 'Let me tell you in my very own words what I think of that: Frankly my dear, I don't give a damn' " ...