Without the Panama Canal, ships going from the east coast of the United States to the west coast, or vice-versa, would have to sail all the way around the southern tip of South America. Taking the shortcut through the Panama Canal saves thousands of miles and days of sailing time.
There are 3 capitals close to the coast, and 2 farther away: --- San Salvador (El Salvador) Managua (Nicaragua) Panama City (Panama) --- Guatemala City (Guatemala) San Jose (Costa Rica) Tegucigalpa is away from the coast, and Belmopan is on the Caribbean side.
A ship wishing to go from the east coast of the US to the west coast would have to go around South America, making the journey a lot longer. Panama wouldn't get the fees they now receive from each ship passing the Canal.
None. (Well, arguably the interior of Panama itself). Any region "connected" with the US east coast via the canal was already "connected" via the Straits of Magellan. The canal did cut the sailing distance (and therefore the trip time) to the US west coast approximately in half, though.