I can only really help with RN uniforms, which might not be what you're looking for. That said, in answer to your questions:

1. The ship's captain wore the same uniform as any other officer of their rank. They wouldn't have a badge to distinguish themselves from an officer of the same rank who didn't command a ship. It's important to note the difference between a ship's captain, and an officer with the rank of captain. A ship's captain was the officer commanding that ship, and had no special badge. An officer with the rank of captain didn't necessarily have to command a ship, and would have a rank insignia of four rings of gold lace on his jacket cuffs.

2. RN officers had white tropical uniforms, rather than khaki. These tropical uniforms were worn on warm-weather stations, such as the Indian Ocean, South Pacific, or in the Mediterranean in summer. Blues were worn on temperate stations - home waters, and the winter Mediterranean. With the exception of the Mediterranean, blues or whites were worn throughout the year.

3. Sailors would change into their working rig when necessary. Those who did dirty work as a matter of course, such as the engine room crew, would wear the working rig day-in day-out. Otherwise, sailors would change into it immediately before doing dirty jobs, like repainting, or repairing machinery.

4. There were vast differences in uniform between ships. It tended to depend on the size of ship, and what the ship was doing. On the larger capital ships, uniforms tended to be policed far more than they would be on the smaller escorts. On the escorts, uniforms tended to be reserved for formal occasions, with both officers and crews opting to wear whatever kept them warm and dry. However, even on the largest ships, the weather was a frequent equaliser between officers and crew - on the freezing Arctic convoy runs, it wouldn't be uncommon for both officers and men to appear out of uniform, bundled up in warm clothing. British merchant ships were generally run by the Merchant Navy, a separate organisation to the RN. This had a uniform for its officers, but Merchant Navy crews could wear whatever they liked.

Real quick. Are you just talking U.S. Navy, Royal Navy, French, German, Imperial Japanese?

Thanks to my google fu, I have found you an actual copy of the 1941 U.S. Navy Uniform Regulations. Navy uniforms aren't as heavily documented as Army ones online, so this took some looking. I also found a history book about US Navy uniforms from 1900 to 1967.

Based upon the review of the regs I can try to answer a few.

1. All officers including the captain had the same uniforms when on duty when underway. The working uniform as proscribed in the regulation were the navy kahkis for comissioned,warrant, and petty officers (NCO's) as well as dungarees. These were worn when in a role where a uniform would not be soiled such as the engine room, maintenance, etc. Yes, this would include the button up coat, but at the commanders discretion in warm climates, the coat could be removed as long as their rank was pinned to their shirt. Officers also had dungarees, but were these were more often worn by the petty officers and enlisted as they were more likely to be involved in the manual labor necessary on ship.

2.) The "blue coat" was a formal uniform that came in the Uniform A, Uniform B, and Uniform C classifications and were used for formal ceremonies as varied as Captain's Masts (punishment hearings), formal dinners, first meetings with foreign officers, etc. The only time it would be worn at sea would possibly be when making a formal boarding-call with foreigners.

3.) Dungarees were the "dirty" standard work uniform for sailors, so any time they were underway or in isolated anchorages thats what they would wear. Dungarees are the navy equivalent of army battle dress (think infantry khakis). These would include a "jumper" which was essentially an outer jacket that was made of the same material as the pants. During the war, since you never knew when you would see action, these became the common uniform.

4.) The uniform was pretty standard across the board...on board, though exceptions were applied for aviators and aircrew and submarine crews. Sub crews were issued waterproofed cotton clothes and other specialized uniforms. There is the special case of aircraft carrier crewmen who would wear various colors (The Rainbow) based upon their various tasks such as fueling, ordinance, catapult and arresting cable crew, crew chiefs, maintenance, flight controllers, etc. As far as I understand the colors haven't changed since WW2.