I can only answer the first half of the question and offer an educated guess to the second half.
For the specific Just-After-Revolution period, we're really going to have to look first at the Russian Post of the Russian Empire. We'll discuss how that system was set up, how robust it was, and then how it was brought into the Soviet fold. Since the history isn't incredibly dense, let's just start with the origin of official post.
#The Russian Empire's Russian Post.
The main Russian postal service has the same name today as it did back during the reign of Peter the Great; "Russian Post". It was during his reign, in 1714, that the first general offices for post were established. This first wave of offices were only in Moscow, St. Petersburg, and Riga (now Latvia). Routes were run every two weeks at first, but the system became more robust and regular by 1720.
This first Russian Post was mostly to ensure quick communication for the Empire and Imperial functions in Moscow and St. Petersburg. It wasn't until well into the 1740s that private correspondence was delivered by Russian Post.
Throughout the 1800s the system became much more robust, but still focused on the cities of St. Petersburg and Moscow. St. Petersburg had 17 postal districts, 42 mail offices, with delivery offices in the suburbs. Mail boxes were installed on the street, near railway stations, coaches, cabriolets, and T-carts delivered larger parcels.
In 1902, Russian Post was rolled into the Ministry of Internal Affairs.
#First Phase of Early Soviet Post (Soviet Russia)
There is quite a gap in deeper understanding of the postal system from 1917 to 1921. I don't think this is due to mail stopping, though it is certainly possible. I think it is mostly due to a few hiccups and disorganized events within the government as a whole.
1). The People's Commissariat for Posts and Telegraphs of Russia was appointed to be in charge of, you guessed it, post and telegraphs. It is said they kept the same organization as the preceding Russian Post, but there appeared to be some confusion about who was in charge of what during this time. On 9 November 1917, the Commissar issued the decree:
I declare that no so-called initiatory groups or committees for the administration of the department of Posts and Telegraphs can usurp the functions belonging to the central power and to me as People's Commissar.
This was a dissolution of any other postal administrative bodies, but how that affected delivery at this time I don't have an answer to.
2). As with all things under the Soviet, censorship was abound. There is a great probability of mail being screened for anti-Communist messages. There's a good likelihood, though not a historical certainty, that people stopped sending messages through the mail out of fear.
3). The wars had crippled telegraph communications throughout Russia which likely in turn crippled the speed and coverage of the postal service.
The fact is I just don't have a good historical answer for you on anything between 1917 and 1921; only some ideas. But I do have more information after that.
#Second Phase Early Soviet Post (Soviet Union)
In December 1921, Soviet Post returned to the history books. The Commissariat began printing and selling semi-postal stamps. These were issued in part to raise money for famine relief. In 1922 these stamps raised nearly 3 million rubles; though the vast majority of such sales came from Germany, not Russia.
During the early 1920s the Soviets began using airplanes to transport mail and in 1923 the Commissariat was rolled into an identically named department responsible for the entire Soviet Union (established in 1922) rather than Russia alone.
By 1925, the Commissariat boasted that nearly 70% of the Soviet populace was covered by home delivery of mail and a rural mobile postal service. It was not until 1929 that the damage done to telegraph networks had been repaired to pre-Great War levels and fax services entered use.
#Educated Guesses and Assessments
What the 1925 boast of the Commissariat tells us is that rural delivery was not nearly as robust as the Commissariat wanted people to believe. Less than a third of Russia's actual population centers would be covered, since the majority of Soviet citizens lived in St. Petersburg or Moscow. The damage to telegraph lines from the Great War and Russian Civil War would have crippled communications for non-state functions. The Soviet control on public discourse would have further constricted what was being sent, by who, to who.
The stamp sales indicate that Russian Post under the Commissariat was still respected to some extent, but the vast majority of these sales were coming from outside the lands of the Soviet Union.
#Conclusion/Addition
The postal infrastructure was robust, but the Great War and Russian Civil War crippled a great deal of it. Recovery was slow and probably exaggerated, but the recovery did happen eventually. From 1917-1921 I'd estimate your odds of successful delivery, as a lay person without connections and, God forbid, anything anti-Communist contained within, were fairly lower than surrounding years. From 1922-1929 your odds were much higher, but contingent on where they lived. If you wanted to mail something out to a remote location without telegraph communications, chance of successful delivery, or the office even accepting the parcel, were very low. If you and the receiver were in urban areas such as Moscow of St. Petersburg, then I'd say those odds became much closer to certain.
There is a lot more information about its growth in the 1930s and recovery from World War II, if you have any interest in that subject.
#References
Bolsheviks and Workers' Control by Maurice Brinton
The Great Soviet Encyclopedia by the Soviet Union