The cold viruses can reach the nose when you rub your eyes because the virus can be passed down the tear ducts that go from the eyes into the nasal cavities. In a word - no. Some people may sneeze if their skin is cooled. More people catch colds when the weather temperature is cold than when it is warm outside because they tend to be inside more often and longer. People tend to blame cool temperatures for getting a cold rather than being in closer, prolonged contact with people who have a cold.
Some people associate exposure to the cool air from air conditioners as a cause of colds.
Again, it is not the temperature that is the culprit. Air conditioners remove moisture from the air which, in turn, can dry the mucous on the nasal membrane. Without a sufficient mucous layer covering these membranes, the nose becomes more susceptible to viruses that cause the common cold. People are most infective or more likely to be able to pass on a cold around two days after they were infected. People can cough or sneeze out in droplets of mucus containing the cold virus.