What? Sweden did not send troops.

All neutral countries that were invaded were invaded by the Germans (Norway only narrowly though, as the Allied invasion force already was on its way when the Germans invaded) because the Germans had strategic aims that required them to invade said countries.

Let's go through the minor nations of Europe and why they were involved.

Neutrals invaded

Poland: Invaded by the Germans 1st of September 1939 since the Germans wanted to reclaim Danzig, the Polish corridor and other territory lost to Poland 1919.

Denmark: Invaded by the Germans 9th of April 1940 as it was required as a springboard for the invasion of Norway. The Danish government and army were allowed to continue to exist until 1943, when they was dissolved amongst strikes and sabotage against German interests in Denmark.

Norway: Invaded by the Germans 9th of April 1940. Wanting to secure submarine bases, the ore trade from Sweden and believing that the British might be on their way, the Germans invaded Norway.

Belgium: Invaded by the Germans 10th of May 1940 as a stepping stone towards France.

Luxembourg: Invaded by the Germans 10th of May 1940 as a stepping stone towards France.

The Netherlands: Invaded by the Germans 10th of May 1940 to secure the flank and supply lines of the invasion of France.

Greece: Invaded by the Italians 28th of October 1940, mostly because Mussolini wanted to increase his influence on the Balkans and the Germans had stationed troops in Romania without informing Italy.

Yugoslavia: Invaded by the Germans, Italians, Hungarians and Bulgarians 6th of April 1941 after a British-sponsored (and widely popular) coup had replaced the government that had signed a transit agreement to allow German troops to attack Greece.

German allies

Finland had been attacked by the Soviet Union on the 30th of November 1939, and had been provoced by the Soviets for a re-match during Spring and Summer 1940. Talks of a state union with Sweden failed, and the Germans offered arms and German troops to help protect Finland. The Finns accepted. When the Germans attacked the Soviets from Finnish soil in Operation Barbarossa on the 22nd of June 1941, the Soviets attacked Finnish troops on the 25th of June 1941, and Finland joined the war against the Soviet Union.

Slovakia: A German client-state since the dissolution of Czechoslovakia 15th of March 1939.

Romania: Joined the axis powers on the 23rd of November 1940, after a coup which placed authoritarian rightwingers in the government after the previous government had given away Dobrudja to Bulgaria, Moldavia to the Soviets and parts of Transylvania to Hungary (all of which was supported by the Germans).

Hungary: Fought Slovakia for the southern part of the country and Ruthenia 23rd of March 1939, then joined Germany as an axis power 20th of November 1940, to get support for their territorial claims. Then partook in the invasion of the Soviet Union.

Bulgaria: Joined the axis powers to partake in the invasion of Yugoslavia and Greece in 1941.

Croatia: Was created as a German puppet state after the invasion of Yugoslavia 1941. Send a reinforced volunteer regiment to the Eastern Front (which fought at Stalingrad) and fought Tito's partisans and the Chetnicks (Yugoslav royalist partisans who eventually switched sides).

Neutrals not invaded

Spain and Germany held talks about a Spanish entry into the war. However, Spain was devastated after the Civil War and Franco knew Spain quickly would lose its remaining colonies to the British in case of an entrance into the war. Thus he demanded huge amounts of supplies, food, raw materials and weapons for Spain to join the war. Spain sent a volunteer division to the Eastern Front (which was a convenient way for Franco to get rid of some of the most fervent fascists in Spain) and exported vital raw materials such as tungsten to Germany, and covertly allowed German submarines to use their ports. Spain did sell the Germans all they needed from Spain, and still had a decently large army, and was a friendly if stingy country, so it was never invaded.

Portugal was under the rule of a right-wing autocratic government under Salazar, but showed no interest at partaking in the war. The Japanese actually invaded Portuguese Timor, but the meagre Portuguese garrison did not offer resistance. Portugal was also a supplier of vital tungsten to Germany, and an economic war was conducted in which the allies tried to buy as much tungsten as possible to deny it to the Germans, and the Germans of course doing the same thing. Portugal saw which way things were leaning and gave the British and Americans basing rights on the Azores (which was important for anti-submarine warfare) in exchange for modern arms. Without control of Spain, it was not possible for the Germans to invade Portugal.

Turkey was under a dictatorship, and benefitted from all powers trying to woo her to their side. France, Britain, the Soviet Union and Germany all sold arms to Turkey at reduced prices. Turkey did sell vital chrome (needed to produce armour plate) to Germany, but did declare war in 1945, without comitting any troops. Turkey had a decently large army, arms and promises of support from many nations and was placed far off. As the Germans mostly got what they needed from Turkey, there was no reason to invade.

Sweden was a democracy dominated by the social democratic party. Sweden sold wood pulp, paper, lumber, ball bearings and above all iron ore to Germany. Sweden was dependent on German coal. Sweden provided massive aid to Finland during the winter war, including a reinforced brigade of 8 250 men of volunteers, entirely equipped and supplied by Sweden. The Swedish army was weak 1940 when Norway and Denmark fell to the Germans, and Sweden had to agree to several degrading concessions to the Germans, including the right for non-armed German troops to travel on Swedish railroads to and from Norway for their leave. A one-time transport of a fully armed German division from Norway to Finland was also allowed. As the war progressed and turned in allied favour, Sweden reduced and eliminated these concessions and stopped selling iron ore to Germany. Sweden also trained Norwegian and Danish refugees as "Police" (in reality regular infantry with HMGs, LMGs, SMGs, rifles, AA guns, mortars etc) and these units were allowed to return to their home countries when the Germans in Norway and Denmark surrendered. Sweden had plans to attack the Germans in Norway and Denmark should they not have surrendered. The Germans did throw their weight around with Sweden quite a bit, but got more dependent on Swedish ball bearings as the war progressed. As the Swedish army got stronger and the Germans had more and more problems, the likelyhood that the Germans would be able to drag men and weapons from the Eastern Front to invade Sweden became smaller and smaller. Early war, the Germans got all they wanted from Sweden, late war, they would have had to weaken a front more than they wanted to invade.

Switzerland supplied the Germans with arms and a banking establishment. The Swiss themselves were not too impressed with the German claim that all German-speakers should be united under one nation, as their confederation worked quite well. The unwillingness of the German-speaking Swiss to be Germans annoyed the nazis a lot. The Swiss army was strong, as were their defence plans and fortifications, while the strategic resources available were small to non-existent. Invading Switzerland would most likely have been too costly to be worth the effort.

Ireland: While there were some resentment towards the British, many Irish joined the British army or worked in British factories and British mines during the war. While the Germans made some secret attempts at negotiating with the Irish, they knew they would not stand a chance against the British and that any German support would be blocked by the Royal Navy.

Switzerland's survival came down almost entirely to the fact that Hitler felt other things were more important. Terrain and troop numbers played a minimal role in the decision not to invade Switzerland.

Some Background:

Hitler hated Swtizerland, even though they were German, they had been corrupted by democracy and he despised the country for its strict neutrality. He lovingly referred to Switzerland as the "Pimple" on the face of Europe. After the invasion of France and the Low Countries, Switzerland was to be next on the list. Field Marshal Leeb was set to head the invasion and he began studying the terrain and planning the invasion. However, other things came up and the operation was put on hold (no one is quite sure what events caused the postponement, most likely Britain's stout resistance, but its open to debate) until Germany could defeat its other more threatening enemies. Since this never happened Switzerland survived.

Source:

A World At Arms by Gerhard Weinberg.

I've written about Spain in the past, so you may find this piece interesting.

In a nutshell though, Spain was devastated by its Civil War still. Franco knew he owed the Axis powers big-time for their assistance, but was always looking out for Spain's interests first and foremost. Although the Blue Division volunteers were allowed to go serve in the East, he made sure to avoid antagonism of the Western Allies, and the Blue Division withdrew once pressure - both allied and domestic - became to much.

In dealing with Germany, Franco did make an offer to help Germany in 1940, but it was rejected, as he asked for huge concessions to join the Axis powers. Germany didn't think they needed the help. Later, when things weren't looking so good for the Axis, Franco rejected essentially the same offer he himself had made in 1940.