There are several common causes of this problem. The most common cause would be an enlargement of a lymph node in the arm pit. The body will shunt any germ that enters the body to the regional lymph nodes, and the germ does not need to have entered by an infection. For example, a small cut, scrape, or bug bite to the hand or arm could allow a germ to enter under the skin. The lymph node may become enlarged as the immune system kills the germ, referred to as reactive lymphadenopathy. It is also possible that the germ can cause an infection in the lymph node, referred to as adenitis. Reactive lymphadenopathy can be slightly tender, but an infection is more worrisome if there is a greater amount of pain or tenderness. It is also possible that it could be a lump within the skin, such as an obstructed sweat gland or a sebaceous cyst. Any lump also could be an abnormal growth, but this is far less common and is usually only a concern if the lump persists for a significant period of time. If it is only a little tender, then it is reasonable to give it a few days to see if it resolves spontaneously. It may help to apply warm compresses to the area. If it persists, becomes more painful, or starts developing other signs of infection, such as redness, purulent drainage, or fever, then you should be seen.
A small amount of redness can occur from inflammation, but if truly reddened is concerning for infection. Another lump would make it more likely to be a problem with a lymph node, as there are numerous lymph nodes in the arm pit.
Yes, unless the other symptoms develop.