Well, this limbo is rather normal period for any country that become independent by declaring splinting from another country. Albeit, from Lithuanian point-of-view was much harder and longer than this one year: Soviet/Russian army remained in Lithuania until 1993, economy simply collapsed due to unsustainability, organized crime surged etc.- essentially large portion of the 90's was an intermediary mess.

Now to your questions:

international community didn't recognize it until September 1991

mostly didn't recognized. Iceland and Denmark recognized Lithuania in February of 1991, and waterfall of recognition began in August, then over 30 countries (most of Europe) recognized Lithuanian independence.

I'm pretty sure that the ethnic Russians were not given Lithuanian citizenship, and thus would have still identified as Soviet subjects.

No,differently than in Latvia and Estonia all residents of Lithuania SSR were given citizenship, mostly because they were much less of the treat: Latvians make up about 60% of Latvia population with majority of the rest being Russian-speaking population (read- soviet colonists), while Lithuanians made over 80% in Lithuania, so there was no real danger in giving citizenship to everyone. How Russian-speaking population felt about Lithuanian independence, however, is entirely another question.

Was capitalism immediately commenced after Lithuania's declaration of independence, or was the country still functioning under a socialist system for some time afterwards?

Privatization program began only in 1991, however, however many enterprises were already falling apart (as testified by this Supreme Soviet of Lithuania document "About economic reform in Lithuania", that tasked government with taking actions against unlawful selling-out and dismantling of various enterprises.

Was Lithuania capable of trading with Western countries without Soviet interference?

No. Thats is the main reason why economic blockade USSR ran against Lithuania in April 20 - July 2 of 1990 was such a big blow and forced Lithuania to declare moratorium of the Act of Independence- Lithuania was tied economically to the USSR and depended on Soviet good will for anything to get in. Given that Lithuanian government did not started regulating foreign trade until May of 1991, much of economic activities in this time period falls in to legally gray area of private initiative and was quite chaotic.

Did Soviet travel restrictions still affect Lithuanians?

Yes. Soviets simply did not recognized independent Lithuania, hence all Soviet institutions worked as if it did not existed. In his „New cold war“ Edward Lucas makes a fond recollection about Algirdas Saudargas, first foreign affairs minister of Lithuania and how helped him to bypass Soviet-controlled border control, for example.

Was Lithuania still part of the Soviet banking system and using the ruble?

Attempts to establish separate monetary system dates back 1989 and Lithuania State Bank was established in March of 1990. Supreme Council officially took over Soviet banking system in October of 1990, however it all ran in to the same problem that Soviets wanted know nothing about this silly nonsense and- long story short- Lithuania used rouble as currency until October of 1992, then temporary currency was introduced only to be replaced by litas in the fallowing year.

Did some Soviet institutions still function in Lithuania even after independence was declared, or was Sajudis able to shut them all down?

Yes again, all Soviet institutions continued to run well in to 1991 an began to dismantle only after August coup. This period is often called – borrowing from Lenin- dvoyevlastiye as Soviet and Lithuanian institutions were kinda just existing next to each other.

What other sovereignty situation could you compare Lithuania in 1990-91 to? Was it a rebellious, self-governing but ultimately illegitimate regional regime like the Confederate States of America?

The best comparison frankly is with other Soviet republics at the time. Your description of CSA is essentially a Soviet point of view, however as the Declaration of Independence was proclaimed by the Supreme Soviet of Lithuania- a parliamentary body elected in February of 1990 in free elections, so question of legitimacy boils down to a question on the view about self-determination: do one recognizes self-determination to be legitimate basis for declaring independence or not.

Was it a "de facto" independent state like Taiwan, almost fully meeting the requirements for a "real country" and qualified only by the international community being reluctant to identify it as truly sovereign due to another powerful country's claim on it?

At least initially it was certainly not, as Lithuanian government controlled nothing and many of participants of the events mention how they half-expected to be arrested almost at any point. Soviets controlled everything and Lithuanian government began to take control of the country only after August coup.

In many ways, Lithuania was a slightly further along example of what was happening in other Soviet Socialist Republics in 1990: all were developing or adjusting their republic-level governmental institutions to mirror changes that Gorbachev was making at the Soviet federal level, and holding elections of various openness to fill governmental positions. Lithuania held elections for its Supreme Soviet in early 1990, and Sajudis (the nationalist independence movement) endorsed a number of independent candidates, as well as some members of the Lithuanian Communist Party (which had already separated itself from the Communist Party of the Soviet Union in 1989). The pro-independence candidates won, and passed the declaration of independence in March.

It's worth noting that the Latvian Supreme Soviet also passed a similar declaration at the same time. The difference was that Lithuania's declaration was to take immediate effect, while the Latvian one declared that there would be a transitional period leading up to de facto independence. Estonia also elected a pro-independence legislature that also began discussions about how to proceed with independence, but did not take any firm steps before the August 1991 coup. (Insert joke about Baltic stereotypes here ... cool, Nordic, pragmatic Estonians, hot-headed Lithuanians, and Latvians in between...).

In some ways, the Baltics were a unique historic case that does not easily fit with the other examples mentioned in the question. The three republics were insistent that they were not declaring independence, but rather restoring de facto the independence that they still maintained de jure. To understand this, we need to go back 50 years.

In 1939, Nazi Germany was preparing for war with Poland. In order to secure plans for invasion, the Nazi Foreign Minister Joachim von Ribbentrop went to Moscow to meet with the Soviet Foreign Minister Vyacheslav Molotov. The two signed a Nazi-Soviet non-aggression pact that stunned the world. However, this treaty contained secret protocols that delineated spheres of influence. Germany would get western Poland and Lithuania, and the USSR would get Estonia, Latvia, Moldavia (then part of Romania), eastern Poland and Finland. A few weeks later, Ribbentrop and Stalin literally [signed] (https://en.wikipedia.org/wiki/Molotov%E2%80%93Ribbentrop_Pact#/media/File:Mapa_2_paktu_Ribbentrop-Mo%C5%82otow.gif) a new map moving Lithuania to the Soviet sphere in return for a few extra chunks of Poland going to Germany. In 1940, the USSR sent ultimatums to the Baltic states requesting that they allow Soviet troops based in their territory, and shortly thereafter the USSR "admitted" them as new SSRs.

I go through all this history, because after 1945, the USSR claimed the three Baltics as member republics of the Soviet Union, but those members of the Baltic states' governments in 1940 that were abroad claimed that no, the three Baltic republics were occupied by the USSR, much as Poland was. The United States and a number of other Western states throughout the Cold War supported this legal position, stating that while the USSR was in de facto possession of the republics, de jure they were still independent states. In a number of Western capitals, the former embassies of the Baltic states were maintained as such and not turned over to Soviet control.

Thus, when the Baltics set about to declare independence, they were in effect stating that it was the Soviet governments and constitutions that were illegitimate, and that it was the pre-1940 states and constitutions that were being restored to their active status. Georgia likewise made a similar statement when nationalists were elected to power there in 1990-1991, declaring the 1921 Soviet invasion to be invalid and that the Republic of Georgia to be restored.

While that is all the theory, the reality is that none of these republics were completely clear and free of control by the Soviet government - renegotiating just who was in charge of what was Gorbachev's main project in 1991, and his scheduled signing of a new Union Treaty (replacing the USSR with a "Union of Sovereign States", with a common federal president, foreign policy and military), was the cause of the failed August 1991, and ultimate dissolution of the USSR.

On the ground, the situation would have been extremely chaotic, much as things were rapidly becoming across the USSR. Soviet institutions were weakening and unsure of their policies, which in any case were being increasingly flouted by ordinary citizens (especially in the Baltics). Gorbachev placed Lithuania under blockade until June 1990, which was lifted in return for negotiations over the republic's final status. While that was ongoing, other countries for intents and purposes still treated the country as part of the USSR, pending a final agreement.

NB: The attack on civilians was on the Vilnius TV Tower, not a radio station. Gorbachev and other members of the Soviet government denied authorizing lethal force, but how responsible they really were is a bit controversial. The clashes ultimately let to a February referendum endorsing independence that was finally recognized after the August coup resolved. It and the other Baltics were admitted to the UN in September, and while a few states had recognized their independence before, most did after the USSR recognized them as independent.