July 12, 1982

For years Dome Petroleum Ltd. was a company that glowed. In the increasingly lucrative oil and natural gas business, it was among the pioneers developing new drilling techniques in the forbidding Beaufort Sea. The stock market applauded, pushing the company's shares to new highs, while people happily referred to its chairman and founder as ''Smiling Jack.'' Today Dome Petroleum is not only in trouble but also is tarnishing the Canadian Government's National Energy Program and making five of the country's major banks extremely edgy about the losses they are risking. The company has historically relied on heavy borrowing to finance its growth, but its debts now are staggering and coming due fast. By the end of 1981, Dome Petroleum's long-term debts were bigger than those of the Ford Motor Company and the Chrysler Corporation combined. As of March 31, Dome reported total borrowing of $5.8 billion (Canadian money) and no cash or short-term deposits whatsoever. A Canadian dollar is now worth about 78 United States cents.

International A fierce artillery fight in west Beirut seemed to indicate that the Palestinian guerrillas besieged by Israeli armored units still had a great deal of ammunition, weapons and resolve. For the first time, the Palestinians fired back on Israeli gun positions in Christian east Beirut. The battle broke up negotiations for a settlement as all diplomatic efforts focused on a cease-fire. (Page A1, Column 6.) Philip C. Habib needs more time to bring about an agreement in Beirut for the withdrawal of Palestinian forces, Defense Secretary Caspar W. Weinberger said. ''A realistic assessment'' of the time needed by Mr. Habib, the special American envoy, is three more weeks, Mr. Weinberger said. Other Administration officials said, however, that they were far from certain that Israel would continue to hold off an invasion of west Beirut for that long. (A6:3-6.)

On the ears of strollers and joggers, bicyclists and motorists, headphones are appearing in force here, and the town fathers have decided that enough is enough. Woodbridge officials, saying they are the first in the nation to do so, have voted an ordinance to limit the wearing of headphones on the town's streets for safety reasons. If the state approves it, the wearing of headphones will be allowed on sidewalks, but when a pedestrian steps over the curb he will have to leave the world of his tiny tape player or radio and return to the sounds of the street. The ordinance also applies to people riding bicycles and driving cars.

Israel is in a quandary about how to get the Palestine Liberation Organization out of Beirut. The political negotiations seem less promising with each passing day. The military option, with its inevitable bloodshed, seems to grow more complex, more costly and less attractive as time drags on. But it also grows more likely, especially as rocket and mortar duels between Palestinian and Israeli forces increase daily. ''The P.L.O. every day raises new conditions and demands,'' said one official in Jerusalem. ''We get nowhere. We are turning around in a circle. We are going backwards instead of forwards. I think it is only a matter of days until we give up this approach.''

One month after its forces surrendered in Stanley, Argentina is a changed nation searching for a new direction. The defeat in the Falklands combined with an economic recession, three-digit inflation, the imposition of Government control over a free-market economy, political confusion, international isolation and even the loss in the World Cup last week have left many Argentines demoralized about their past and questioning the future. A mood of uncertainty hangs over the country. ''I feel dizzy,'' a businessman said. ''All the traumatic shocks came together at once.'' Gone are the signs of nationalistic euphoria that existed during the war -the rousing television advertisements, the billboards declaring ''Las Malvinas son Argentinas,'' using the Argentine name for the Falklands.