Which faction had tactical edge over another and how did this 'edge' change dynamically over time? For example, I heard that during the early wars, IJN had air superiority with their Zeros but the edge disappeared as as the war progressed and Americans developed better planes.
This is a topic you could probably write a full-length book about. Instead of doing so, I'm going to link to a couple of other posts I've written about this topic. This thread mostly discusses the RN, but does explain how it compares to the other navies of the war. In this thread from /r/WarCollege, I discuss the strengths and weaknesses of most of the major navies in 1939, though I didn't quite finish discussing the USN.
I am going to write about the problems and advantages with the Kriegsmarine and i am going to base most of my answer on James Holland`s "The war in the west:Germany ascendant 1939-1941"
By the outbreak of WW2(1 september 1939) Germany had 4 battleships(of which 2 were older outdated ships)+11 under construction, but only Bismarck was completed, 3 pocket battleships(well armed, fast, but lacked the armour of battleships. No aircraft carriers, 2 heavy cruisers+1 under construction, 6 light cruisers, 22 destroyers+12 under construction. They also had 20 torpedo boats and 13 being built and 62 submarines with 50 new being built.
Compared with the British and French fleet, this shows an extreme lack of capital ships. The two allies had a total of 22 Battleships(BB), excluding the 13 under construction, 8 aircraft carriers(CV) with 6 new being built, 22 heavy cruisers and vastly superior numbers in smaller ships(Britain alone had something like 180 destroyers(DD) alone, with the french having almost 80, pluss corvetes, submarines, minesweepers and gun monitors). This lack of capital ships made it impossible to fight the Allies in a decisive naval battle and they therefor deployed their fleet as convoy raiders. Due to this the most important part of the Kriegsmarine was the submarines lead by Karl Dönitz. Dönitz recognised that the goal of the German fleet should be to starve the Great Britain from their all important trade and imports and to achieve this he advocated for a focus on submarines. Sadly for Dönitz the Oberkommando der Marine(OBM) decided to start on Plan Z, an ambitious plan to expand the german fleet. This lead to Germany using the years before the war to build an expensive surface fleet that in the end failed to challenge the Royal Navy. Although Plan Z included some submarines the Kriefsmarine had way to few Submarines at the start of the war to sink enough shipping to starve Britain and therfore neither the submarines nor the surface fleet could achieve enough for germany to win the war. Although there were to few submarines in the Kriegsmarine, the crews were highly trained and very efficient. There were also some problems with how the submarines were used. For example, during the Norway Campaign, Dönitz were forced to deploy a number of submarines against his will, arguing that they were far better used in the Atlantic hunting merchant shipping. An other example is the forced deployment of multiple submarines north of Norway at the beginning of Barbarossa. This was done despite the fact that no shipping had been sent that way, making the submarines usless.
To sumarize, the main problems were the lack of a surface fleet capable of challenging the britsh fleet and not enough submarines to bring Great Britain to it`s knees through sinking their shipping.
This is not a complete answer, and there is more to be said, but this covers some of the important aspects. If anyone finds something wrong, i would be glad to be corrected and i would also be glad to answer any further questions.