Red Spot Beside Mole?

I would recommend that you see your physician for an evaluation and possible treatment.  It may be that you have an acne cyst next to the mole, but best to check with your dermatologist.

This could be nothing but a blood vessel. But these changes or charateristics are more worrisome than if they weren't there. You might want to consider complete removal and a pathological diagnosis.

You should have your mole examined by a dermatologist.  You may just have a recurring acne cyst beside your mole and it can also be treated by the dermatologist.

It is impossible to tell you what is going on without seeing the lesion.  You would be best served by calling a local board-certified dermatologist.  The American Academy of Dermatology (aad.org) has a list where you can find one. 

Whenever a mole is changing it is important to have it checked by a board certified dermatologist. Please use the ABCDE's as your guide for when to be concerned about moles: A for Asymmetry; B for irregular Borders; C for black or uneven Colors; D for Diameter greater than a pencil eraser; and E for a growth that is Evolving or changing. It is an extremely simple five minute procedure for a dermatologist to remove the mole using either a "punch" or a "shave" biopsy which typically leaves a minimal scar. The mole can then be checked microscopically for reassurance.

You can follow the video link below to learn more about identifying the changes in moles that are suspicious for skin cancer.