The proposed joint mission between both NASA's Apollo and the Soviet Soyuz is a homage to the Apollo-Soyuz test Project, the first ever joint space mission between two countries which actually took place in July 1975 with NASA's Thomas P. Stafford, Vance D. Brand, Deke Slayton and USSR's Valeri Kubasov and Alexi Leonov. The mission was a cultural success and Stafford and Leonov became friends and remained so until Leonov's death in 2019.
One of the Soviet cosmonauts pays tribute to Laika, the first mammal in space aboard Sputnik 2 in October 1957. The spacecraft did indeed fail to separate and this created problems with the thermal control system. Laika died from the stress within a few hours but this was not widely known in the West until 2002. Previously Soviet officials stated she was euthanized on day 6 of the mission.
It's suggested in jest that the missed shot may end up going all the way around the Moon. The M16 rifles depicted in the movie fire a 5.56x45mm round at about 900m/s, a bullet would need to be going about 1700m/s to achieve orbital velocity.