Why does crying lead to a running nose and reddened nose tip?
When you cry, you produce tears, which come out of your tear ducts. These are not only open to the eyes but also have a connection to your nose, see the image below:

 

A part of your tears runs through this connection and mixes with the mucus which is present in your nasal system making it thinner and allow it to flow more easily. If the tear flow is prolonged, the nose will produce more mucus and make your nose running for a longer time. The nose appears redder because the blood flow is increased. This is partly due to the production of more mucus and also because you repeatedly touch the nose when using a handkerchief.