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.
Sources:
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)
Each country will, of course, have its own history, and more can always be said even for these ones, but some of this has been covered in previous answers on the subreddit. So while we wait for more to weigh in, I would point to this answer by /u/commiespaceinvader which focused mostly on Monaco, but does also touch on Andorra and San Marino.
The relationship specifically of the Vatican to the rest of Europe during World War II is complicated, often politicized, and has been highly controversial. A WWII historian may be better able to describe the day-to-day or more temporal political situation in the Vatican at the time, but I can give a broad overview of the Vatican's positions from a broader church policy standpoint.
The Vatican only became the independent state as we understand it today in 1929, when Pope Pius XI and Benito Mussolini signed the Lateran Treaty, acknowledging Papal sovereignty over the Holy See (from the dissolution of the Papal States in 1870 until 1929, the Italian governments and the Papacy were at odds over the administration of Rome, leading to the Popes of the period being referred to as "prisoners in the Vatican"). This legal recognition of independence likely gave the Papacy some greater leeway with regard to policy with respect to WWII than they might otherwise have had, but the Church's hands were still somewhat tied by virtue of their geography. Add into this the fact that majority-Catholic nations fought on both sides in WWII (examples include Poland and France for the Allies and Italy and Croatia for the Axis) and the situation becomes more complicated still.
Pius XI, Pope from 1922 to 1939, signed a treaty (the Reichskonkordat) with the new Nazi government of Germany in 1933, which was intended to lay out the boundaries between the Catholic church in Germany and the government. The Reichskonkordat was heavily criticized by Catholics in France and Poland and by non-Catholics in Germany as the Church throwing itself in with the Nazis. However, even if the treaty was favorable to the Nazis as written, Hitler almost immediately began violating its terms. Persistent violations of the Reichskonkordat led to Pius XI issuing the encyclical Mit brennender Sorge (With burning tears) in 1937, which condemned condemned "pantheistic confusion", "neopaganism", "the so-called myth of race and blood", and the idolizing of the German government. The encyclical precipitated an intensification of the Nazi persecution of the Catholic Church and permanently damaged Nazi-Vatican relations, but has been criticized for not being strong enough (the encyclical never mentions Hitler or the Nazi Party by name, choosing instead to criticize cultural trends associated with the Nazis). Pius XI's other encyclicals dealing with the political situation in Europe at the time, Non abbiamo bisogno (1931) and Divini redemptoris (1937) have also been criticized as inadequate, although they call Catholics to prevent and avoid a "deluge of discord" and "poignant disasters."
It was to this environment that Pius XII was elected as Pope on March 2, 1939, just six months before the German invasion of Poland on September 1. His first encyclical, Summi Pontificatus, published in October 1939. The document condemns the invasion of Poland, racism, and anti-Semitism. Although Summi Pontificatus takes an unambiguously anti-Axis line, and which Pius XII continued to condemn the Nazi regime, he was often silent on Mussolini's cooperation with the Germans and, as the war progressed, tended to refrain from criticizing the Nazis too harshly because he feared German reprisals against Catholics in German-occupied Europe. In spite of this, Pius XII's opposition to Nazism has been praised and he has been referred to as "a lonely voice crying out of the silence of a continent."
While the Papacy took a harder line towards the non-Catholic Nazi government and their persecution of Catholics in Germany, Poland, and Czachoslovakia; it did not take as strong of a position towards the Catholic Ustaše regime of Ante Pavelić in Croatia. Pius XII met with in Pavelić 1941 and did not publicly condemn the expulsions and forced conversions to Catholicism perpetrated on Serbs. However, the Vatican Secretariat of State privately repudiated them in a 1942 memorandum to the Yugoslavian Legation. In 1953, Pius XII would promote Aloysius Stepinac, the Archbishop of Zagreb from 1937 who was convicted of supporting the Ustaše and genocide of Serbs, to a cardinal.
The Vatican during WWII had to navigate a treacherous political situation in Europe, and this was complicated by its geographical situation and its desire to continue to fulfill its religious role with as little political interference as possible while at the same time avoiding schism within the Church and maintaining Christian doctrines against evildoers. The Holy See was notable in its condemnation of Nazi atrocities and in its hearty condemnation of anti-Semitism; its attitude towards atrocities committed by Catholic regimes against non-Catholics was, at best, to privately and quietly disapprove.
Finally, this is by no means an exhaustive answer with regard to church policy during WWII. Such an overview can and has filled multiple books and has been subject to much scholarly disagreement. I have tried to provide as objective and unbiased of an answer as I can, but I am aware that true objectivity is impossible and I welcome the response of other commentors who may hev differing viewpoints.
-- Some helpful literature (in no particular order and once again by no means exhaustive):
An interesting primary source is:
Thank you for all the interesting answers. As a follow up: How was life in these countries? Did the people live in constant fear? How did they trade? How did they get food and fuel?