Ok, probably the most important of these within the scope of WWII was Monaco.
Monaco was similar to Switzerland know as a country with very advantageous tax and financial laws and also akin to Switzerland it featured prominently in the scheme of Nazi Germany to use neutral countries with such laws in order to gain access to highly coveted foreign currency, which was one of the resources the Third Reich lacked massively in. Essentially buying and selling goods and currency through shell companies, Monaco was very important to several schemes by the German finance ministry and its task of supplying the war build up with foreign currency.
This was one of the major reasons that with the fall of France in June 1940 to German troops, Monaco was not occupied by the Germans. It was however, occupied by the Italians, who at this point had declared war on France too and when marching up the Mediterranean cost occupied Monaco much to the chagrin of their German allies. Due to German pressure, the Italians retreaded from Monaco and the ruling Prince Luis II. initiated a policy of friendly neutrality towards the Axis put in practice by praising Vichy France and in July 1941 issuing a law requiring all Jews who had fled Germany occupied France to Monaco to register. While some of them were thrown out of Monaco and there are a couple of cases of people being deported to concentration camps on pressure of the Germans, no systematic policy came of it however.
In November 1942 following the Allied landings in North Africa, Italian troops again occupied Monaco and installed a fascist puppet regime. This occupation was in turn ousted by a German occupation after Mussolini was deposed in September 1943, which lasted until September 1944 when Monaco was liberated by Allied troops. One of the major reasons Monaco was able to remain independent despite Louis II. appearing as a collaborationist regime to many in the Allied political echelons was the the heir to the throne was declared to be the nephew of Louis, Rainier in May 1944. Rainier had joined the Free French Army after Monaco’s liberation in September 1944 and subsequently took part in the liberation of the Alsace and received the American Bronze Star Medal for his actions during that campaign, making him one of the few governmental heads in Europe to actively take part in the fighting.
By the outbreak of WWII, San Marino had been ruled by the local Fascist Party since 1923 but despite that remained neutral in the conflict. In September 1940 it was even reported that San Marino had declared war on Great Britain, a mistake it was quick to clear up. Through the deposition of Mussolini and the subsequent German occupation of Italy, San Marino was also affected not at least by the collapse of the rule of the Fascist Party there. In September 1944, it was briefly occupied by the Germans and also sight of a major battle between Allied and German forces. Interestingly enough, after having been under fascist rule during the war and German occupation, San Marino also saw Europe’s first post-war democratically elected communist government, which in coalition with the Social Democrats ruled from 1945 to 1957.
I’m afraid there is even less to be said for Andorra. It had been de facto occupied by France following social unrest in 1933 and remained so to «protect» it from disruptions from the Spanish Civil War. During WWII it remained neutral despite due to having been forgotten in the Versailles Treaty after having declared war on Germany in the First World War still being officially at war with Nazi Germany. The main impact of it was that the French Resistance used it as a base to smuggle downed Allied airmen out of France.
As for the rest of your question, I am unsure what you’d deem as «small». Next to the Vatican, which has its own history upon which you’ll also find in the FAQ, is Lichtenstein, which has a similar history to Monaco or Switzerland in the conflict, mainly serving as a hub for German financial activities. The Nazi Party of Lichtenstein did try to topple the prince of Lichtenstein in 1939 but was unsuccessful in doing so probably due to the fact that it only had about 25 members.
Peter Geiger: Krisenzeit: Liechtenstein in den Dreissigerjahren, 1928-1939, Vaduz: Historischer Verein für das Fürstentum Liechtenstein; Zürich: Chronos, 2000.
Bettina Grosse de Cosnac: Die Grimaldis: Geschichte und Gegenwart der Fürstenfamilie von Monaco. (pop history and kind of bad, I don’t even know why I have this one around)
Dieter Langewiesche: Kleinstaaten in Europa: Symposium am Liechtenstein-Institut zum Jubiläum 200 Jahre Souveränität Fürstentum Liechtenstein 1806-2006 (this one was my main source, I am sorry I am not able to offer something up in English for you but as you can imagine this is a rather obscure subject. I would generally try to find out more about San Marino and Monaco from WWII histories of France and Italy)
Мысль на память: Можете отчитаться за каждый заработанный миллион, кроме первого.