I shall attempt to give some perspective on the first question you have posited here. The "too long didn't read" response would simply be: Nazi Germany could not govern those countries' overseas colonies. Though the "mother countries" of France, Belgium and the Netherlands formally surrendered to the Germans, their overseas colonies were not generally included as clauses within the respective treaties.

The main exception to this trend of course was France, where the Vichy regime allowed Nazi Germany limited political control over its North African territories (including French Algeria and Morocco), though this mainly constituted of a joint-government between Nazi Germany and Vichy France. Primary accounts of the time are a tad hard to come by, but it seems as though the Germans were content to allow Petain's regime to handle most of the day-to-day business; they of course, perhaps rightly so, were more concerned with their Afrikakorps fighting within a few hundred miles of the territory.

Interestingly, Madagascar as a French Colony actually saw an Allied (mostly British) invasion in 1942 in order to "deter" Japanese-German plans in the area. The invasion sustained minor casualties, though on another tangent Madagascar was under consideration as an early"Final Solution"; Franz Rademacher (head of the Jewish Department of the German Foreign Office) had suggested to Hitler and Himmler that they deport all of Europe's Jews to the remote island, but this idea was scrapped after the Battle of Britain started (mostly due to the British naval blockade in the North Sea but also due to the Wannsee Conference the following year.)

With the Dutch the situation was fairly similar though the Germans were even less involved: the Dutch West Indies were not even a primary consideration of them or the Allies and their role in the war is (perhaps unjustly) ignored even more than the other colonies. On the other side of that "recognition spectrum", the Dutch East Indies were very much involved in the war. Germany, Italy and Japan had agreed that this state would fall under the Japanese sphere in 1940, and in 1942 the Dutch East Indies campaign successfully forced the island governor's surrender of the colony by March. For the next three years, the Japanese occupation of the island would be described by many contemporary Indonesians as "harsher and more oppressive than the Dutch one had been", a major factor in the fledgling independence movement in the area.

The situation with Germany was much the same for the Belgian Congo and other colonies in Africa: limited gains and none that were particularly helpful The British and French were quick to secure these territories from their neighbouring colonial settlements in Africa. The resources of the Congo were particularly precious and useful for the American wartime industry: copper, rubber and various minerals were extracted and shipped over the Atlantic for transformation into equipment. Most notably however, were the Shinkolobwe mines in the area, which infamously provided the uranium required by the Manhattan Project (and rather sadly, would be a main reason why the country was so hotly contested by the USA and USSR in the Cold War to follow).

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In conclusion from those fairly sized explanations, the Germans were unable to gain significantly more than the European territories and related assets from the Low Countries and France. Only France (and its Vichy collaborationist government) made concessions (and even then the Germans were not in a rush to dominate the North African colonies), the rest were either "saved" by the Allies or left to fend for themselves (in the case of the Dutch East Indies) against the Japanese.

This is because the Third Reich did not take over those colonies. As Starwarsnerd2222 posted, the situation for the French colonies is more complicated, but the Dutch colonies fought on. General Winkelman's surrender of the troops in the Netherlands on the 14th of May 1940 didn't even include the province of Zeeland, which was taken on the 17th.

The experience in the Dutch West Indies and Suriname was a lot different than those in the East Indies After the occupation of the Netherlands, it did not have a lot of forces left for its own defence. Their importance to the Allies came from the oil industry on Aruba and Curacao and the bauxite coming out of Suriname. The Dutch West Indies were chiefly secured by American troops during the war, who dwarfed the local Dutch garrison.

The experience in the Netherlands East Indies (NEI) can be divided into three parts: 1. from May 1940 when the homeland was occupied until the war broke out against Japan in December 1941, 2. The war itself, which lasted from December 1941 until the occupation of Java in March 1942 and 3. Dutch forces exiled to Ceylon/Sri Lanka and Australia until the end of the war in August 1945.

1. In this period the NEI tried to improve its own defences, patrols against (suspected) German raiders and postures against an more aggressive Japan. The European population did feel a connection with the war in Europe: there were several initiatives such as collecting money to buy planes for Dutch squadrons in the RAF.
2. Allied with American, British and Australian forces attempting to defend the islands. Dutch strategy, having inferior forces, was relying on using the Navy and Army Air Services to find Japanese forces and then directing submarines onto the line of advance of those forces. Surface forces were to try to achieve local superiority of numbers and then engage. However, this was hampered by Japanese air control, lack of coordination between allies and experience in working together (official talks only started after December 1941) and having large amounts of Dutch forces allocated to the defence of Singapore instead of the NEI.
3. Being driven out of the NEI, Dutch forces regrouped in Australia and Ceylon. The raids on Darwin however, destroyed much of the modern planes of the Naval Air Service evacuated there. Dutch forces were placed under Allies operational commanders and used as part of larger Allied units. Guerilla forces held out for much longer in the interiors, before being found out or evacuated. Attempts to land agents in the Japanese occupied areas were mostly unsuccessful: agents were often given up by locals. Obviously the Dutch population of the Netherlands would take much greater risks to help an anti-German resistance in the Netherlands, than Indonesian people would to help Dutch anti-Japanese Resistance in order to restore Dutch authority.

Most sources I know are sadly Dutch, but they include general histories of the Royal Netherlands Navy during the Second World War and memoires. For instance:

K.W.L. Bezemer Zij vochten op de zeven zeeën, (1954).

M. Bosscher De Koninklijke Marine in de Tweede Wereldoorlog, (1990).

A. Kroese Neerland's zeemacht in oorlog, (1944).

Seeing how most of the RNLN fought in the NEI, the events they describe often take place there. However, the earlier books still talk about the need to reoccupy the NEI for the good of the Netherlands. I would see them in this context as sources on how the RNLN saw its involvement in the war, seeing as they are official histories.

Books such as De Tegenaanval(2016) by H. Bussemaker and J. van Klink and De strijd om de slagkruisers 1938 - 1940 (1984) by G. Teiler give a primer in the Dutch discussion on how to defend the NEI, while Piet Hagen's Koloniale oorlogen in Indonesië (2018) gives the subject a broader context.

Also interesting would be the Memoires by Admiral C. Helfrich, published in 1950, which I am currently reading. He was Dutch Naval and later Overall Commander in the East.