Eastern European cold war movies

One good example would be "Encounter at the Elbe" (1949) :

The Soviet Red Army and the American army meet at the Elbe, in Germany, thereby ending World War II with a victory. Still, there are new challenges in the Soviet occupied territory in peace, too, namely rebuilding the cities and winning over the sympathies of German civilians. The Soviet commander in charge of the area, Kuzmin, creates a friendship with German scientist Dietrich and persuades him to become the mayor of Altenstadt. However, when Dietrich's patents are stolen, he suspects the Soviets and flees to the American zone. There, he is shocked by the US mismanagement of Germany, and returns humbly back to the Soviet zone.

If I remember it right, it also was showing horrible racism among the US soldiers, with black private beaten up and kicked from a pub.

Another example: "I am Cuba" that shows the contrast between rich Americans and poor Cubans before the Castro revolution:

Farmer Pedro, who just raised his biggest crop of sugar yet. However, his landlord rides up to the farm as he is harvesting his crops and tells him that he has sold the land that Pedro lives on to United Fruit, and Pedro and his family must leave immediately. Pedro asks what about the crops? The landowner says, "you raised them on my land. I'll let you keep the sweat you put into growing them, but that is all," and he rides off. Pedro lies to his children and tells them everything is fine. He gives them all the money he has and tells them to have a fun day in town. After they leave, he sets all of his crops and house on fire. He then dies from the smoke inhalation.

Maria lives in a shanty-town on the edge of Havana and hopes to get married to her fruit-seller boyfriend, Rene. Rene is unaware that she leads an unhappy double-life as "Betty", a bar prostitute at one of the Havana casinos catering to rich Americans. One night, her client asks her if he can see where she lives rather than taking her to his own room. She takes him to her small hovel where she reluctantly undresses. The next morning he tosses her a few dollars and takes her most prized possession, her crucifix necklace. As he is about to leave Rene walks in and sees his ashamed fiancée. The American callously says, "Bye Betty!" as he makes his exit.

In general, in the further movies, West was shown not as much as "evil" but as "corrupted"("Rotten Imperialistic West") - a place where Mafia and ruthless companies rule over everything, black people are treated as second class citizens and culture is reduced either to mindless consumption or heroine fueled "noisy rock'n'roll". Unfortunately, I can't recall many of such movies - most of the propaganda described above was coming from news or documentaries.

Did the Eastern European countries, and particularly Soviet Union, have their own “cold war movies”?

Yes, they did. One of my personal favorites is TASS Is Authorized to Declare...

Some more examples: Dead Season, Long Castle, Meeting the spy, The Secret Agent's Blunder and many many more.

And I can't not to mention Seventeen Moments of Spring - the most popular Soviet spy movie. While the setting is Nazi Germany, the theme of the movie is how the Soviet spy "Stierlitz is tasked with disrupting the negotiations between Karl Wolff and Allen Dulles taking place in Switzerland, aimed at forging a separate peace between Germany and the Western Allies" (the quote from Wikipedia). The main enemy are the nazis of course, but Western Allies aren't very good either.

Mostly no because that topic was cultivated in propaganda division of USSR and soviet satellites countries.
So when in USA you had PSA movies about drunkenness in soviet camp you had a movie about "how bourgeois capitalist forced drinking to an excess on workers' and peasant collective". We had movies about colorado beetle being dropped by American planes on our potatoes. Movies about bad youth? Of course, but the baddines came from rotten east that lure young people with jazz and shoes with fat soles instead of studying and making our socialist country better.

On the other hand regular movie director we're not interested in creating film that were topic related. They didn't had to. Money was not an issue, there was no such thing as profits so they could make artistic movies that were send to shelf by censors.