What were some major military developments in Germany, Italy, England etc. like around the same time?

I would first wonder why you might think the sinking of a single minor gunboat in an area of international conflict would really have changed things between the US and Japan?

The basic answer is that while certain individual decisions and events in the late 30's brought tensions to a head and set the course for war starting really in 1940 and early 1941 even then it was not truly a given until the Fall of 1941.

Nor had it begun only in the mid 30's. From the entrance of the US into the Western Pacific as a true colonial power following the war with Spain in 1898, and Japans emergence as a true imperial power after her victory over Russia in 1905 collision courses were set. Geopolitics ensured that two newly self assured powers would not look kindly on the other trying to carve out the largest sphere of influence in a rich potential market and region with numerous value bale resources. Doubly so when Britain realigned herself from an ally of Japan to that of the US following WW1 when she allowed a naval treaty and other agreements with Japan to lapse and moved closer to her Anglosphere kin.

Add in some good old racism on both sides, which itself nearly threatened war a few times, and you have all the ingredients for conflict. Indeed whole generations of naval and army officers on both sides grew up preparing, thinking about, and playing out wars between the two prior to 1941. The famed War Plan ORANGE for a transatlantic war against Japan was first adopted in 1924-25, but antecedents dated all the way back to 1906 after Japan established itself as a credible naval power. It alternated with the years as a quick thrust to get a fleet to the Philippines to save the islands and set up a base as possible, and a more reasoned buildup and island hopping campaign to establish strong bases and supply lines as the fleet went which informed many of the decisions which would actually be put into practice in the war.

But to get to more the meat of your question after establishing that the powers never stopped looking askance at each other for almost 40 years with a handful of exceptions, we can certainly see a massive spike in innovation, acceleration in defense spending, and rearmament as the 30's wore on. But the Panay incident was simply one data point of increasing tensions. the far great motivators were Japan's imperial adventures in China, and its withdraw and abandonment of International Treaties and groups.

Specifically about 2 years before the Panay's rough day at Nanjing Japan withdrew from negotiations for the 2nd London Naval Treaty in January 1936 which was its hardest break yet with the International order and a strong signal it was taking a new track. Through the 20's and 30's a large series of globally disarmament conferences had been held, especially for naval fleets in an attempt to prevent the dreadnought race that had inflamed tensions between Britain and Germany before WW1. The most successful on in Washington in 1922-23 led to a major reduction with many older warships globally scrapped, and a 'Battleship Holiday' of a decade and a fixing in general tonnage of capital ships of 5:5:3:1.75:1.75 for Britain, the US, Japan, France, and Italy. Later talks at Geneva several times, and a Treaty of London in 1930 extended the holiday to 1936, and extended additional limits to other types of ships like cruisers, destroyers, and subs.

In Japan though this was seen as an abject humiliation by the white colonial powers to conspire against the ascendant Yamato race, at least in some very influential circles. And while the 5:5:3 ratio was barely accepted it divided the fleet and nation and helped fan the flames of militarism which would sink the democratic elements of the Japanese government over the next decade, with a fair few killings and attempted coups by military hardliners and unauthorized expansions of the war in China. All of which of course only increased tension with the West in turn. So in 1934 Japan announced it would not agree to continue the Washington Treaty as it was past its sunset date on 1936, and then withdrew from the 2nd London Conference in early 1936 which was meant to figure out a way to continue it. The remaining powers did agree to a limited treaty which included an escalator clause that allowed them to build battleships with large new 16in guns should Japan or Italy not sign the treaty. Japan also withdrew outright from the League of Nations in 1933 following its conquest of Manchuria.

And while the collapse of the treaty system allowed builders and designers more free reign, even under the treaties the fleets worked to build the best ships they could to defeat their foes. The 'Treaty Cruiser' competitions were especially fierce with ships limited to 8in guns and 10k tons it could be enough for a ship to be good at really 2 of the 3 factors of speedy, hard hitting, or well protected, and squaring that circle was the job of many a naval architect. Japan even before leaving the treaties also tended to play more fast and lose with the restrictions and it was much more difficult for Western intelligence officers to confirm than their other peers.

However if we just restrict ourselves to capital ships of battleships and carriers we see a sharp increase in effort spent, ships designed, and ships built as the treaties collapsed. Japan had begun design studies along its doctrine of heavily armed quality over more survivable quantity for its new class of battleships after it left the LoN. The Yamato class would be the largest battleships to hit the water with the largest guns ever put to sea at 18.1in, and started construction in 1937 after costly expansions of docks and shore facilities. The US also began new constructions of battleships with the 6 new fast battleships of the North Carolina and South Dakota classes using 16in guns under the escalator clause int he rump 2nd London treaty. While much better than the old Standard Type battleships they had to make unsavory trades in sea keeping abilities, armor, internal layout, and were not as fast as might be liked in order to mount the big guns, still fit the Panama Canal, and still be under the old 35k ton weight limit though 45k ton designs were now allowed after the escalation. The follow on Iowa class was a much better mix after using much of that extra tonnage to increase her speed by basically stretching the design out for better lines, though still not perfect. Britain also restarted construction with her King George V design which mounted the old limit of 14in guns in a single dual gun turret and a pair of complex 4 gun turrets.

Similar developments attended aircraft carrier construction with each new generation of aircraft getting closer to turning naval aviation into a truly decisive arm, and the last generation of designs prewar finally doing so. Many of the most famous flattops of the war were constructed as tensions rose in the 30's using new and innovative ideas, Soryu and Hiryu, Shokaku and Zuikaku would form 2/3 of the famed Kido Butai for the Imperial Japanese Navy. And building on the earlier conversions of the coalier Langley, battlecruisers Lexington and Saratoga, and modest new build Ranger the USN sought to fill out her limit of carrier tonnage with a new deign absorbing lessons from them. So in 1934 the famed sisters Yorktown and Enterprise began construction, with the slightly smaller Wasp to follow and true sister Hornet as an easy quick boost in 1939. Similar arcs can be traced in all varieties of ships, and as tension increased so did budget and appropriations. The first and second Vinson Acts in 1934 and 1938 and Two Ocean Navy Act of 1940 funded the design and initial construction of many of the most important ships of the Pacific War for the US including the first batch of Essex class aircraft carriers. Japans Naval Armaments Supplemental Programs took a similar track.

Hope that makes sense, for Europe I would always turn to the fantastic /u/thefourthmaninaboat and if you are interested in some more reading on a specific topic there is always more to look at!