For the scene where the cow is on fire, it was covered in asbestos, which protected it from actually being burned. But for the scene where the horse falls down the stairs, it was shot in the head. The crew acquired the horse from a slaughterhouse, and it was going to be shot the next day, so they decided to use it for the film.
Metal being cast while molten is generally accomplished using a special type of totally dry sand for the mold, not clay. Clay normally contains moisture and should not be used for casting metal because when the metal touches the clay, steam is produced in large volumes, causing the metal to splash about wildly. Evidently neither Tarkovsky nor Konchalovsky (the scriptwriters) knew this, or if the film is historically accurate, then there must have commonly been a lot of injuries or deaths in the Russian bell casting trade in the 15th century.
The character of Danil is based on Daniil Chyorny (c. 1360 - 1430), a Russian icon painter and companion of Andrei Rublyov. He is believed to have painted the Assumption Cathedral in Vladimir in conjunction with Rublyov.
The Andronikov Monastery was located in the Taganka region of Moscow and was built in 1360 on the eastern bank of the Yauza River as part of Moscow's outer defensive ring of monastery-fortresses. Its name is derived from that of its of its first abbot (Andronik). The monastery's most famous monk was Andrei Rublyov.
The movie was completed and shown to selected people in private screenings in the winter of 1966. The first official screening was in February 1969 in Moscow, followed by a screening at the Cannes film festival in May 1969. International distribution started in 1973.
There are two major churches mentioned. Dmitrievsky Cathedral (1194) is located in the city of Vladimir. The Annunciation Cathedral is in Moscow and is an amalgamation of churches and chapels from the 14th to the 16th centuries. It is the second oldest cathedral in the Kremlin.