After the Russian Revolution, and the treaty of Brest-Litovsk, many small nations formed, such as Armenia, Ukraine and Finland. How exactly did these short lived countries operate and did any of them have any real authority?

It of course varied by country. The three Baltic states (Latvia, Lithuania, Estonia) all became independent in the aftermath of the Revolution, and maintained this until 1940 when they were annexed by the Soviets (part of the secret articles of the Molotov-Ribbentrop Pact with Nazi Germany). They did have some Bolshevik insurrection at first, attempts to destabilise them and force them back into Russia, but they were quite independent in this time; Lithuania's biggest concern in this era was actually Poland, who occupied Vilnius (the modern capital of Lithuania).

Ukraine and Belarus both had states formed in this time, though they were more limited I would say. They were heavily aided by the Germans, and Ukraine was further complicated in that it was a literal battleground between multiple forces (Bolsheviks, White Russian forces, Ukrainians, Poles, Germans), and was arguably formed as a German puppet state. It was also dealing with internal struggles, as Western Ukraine (centered on Lviv) had declared its own independence briefly. This was all sorted out finally by the Treaty of Riga in 1921, which ended the Polish-Soviet War; most of modern Ukraine was annexed by Soviet Russia, while Western Ukraine was included into Poland (and ultimately annexed by the USSR in 1939).

In the Caucasus, there were three main states that existed, all in the South (as well as some attempted minor states in the North Caucasus): Armenia, Azerbaijan, Georgia. They initially formed a united republic in 1918, the Transcaucasian Democratic Federative Republic (which was actually proclaimed almost 100 years ago to the day, April 22, 1918), but it only lasted one month, as the three states inside couldn't agree on international affairs (mainly the issue of how to deal with the Turks; Azerbaijan favoured working closer, Armenia not so much, while Georgia was also somewhat reluctant). Georgia declared independence on May 26, while Armenia and Azerbaijan did the following day.

All three of them did exist in a nominal sense. I hesitate to say they were viable, as they did all end up getting invaded and annexed by the Bolsheviks, but they each had working governments, and elections and policies. Security was a major issue for them all: Baku in Azerbaijan was a hotbed of revolutionary activity, and the inability of the government there to quash that saw the Bolsheviks invade it first, in April 1920. Armenia was next, in December 1920, while Georgia held out until March 1921.