What sort of headaches give you pain behind eyes, red patches under eyes & also make them "puffy" ?

Some: Thees can be cluster headaches, much more rare in women, but no unheard of. There is a group of disorders call trigeminal autonomic cephalgias (tac) and hemicrania continue. All of these have some overlapping similarities. Treatment with Indomethacin is a "test" so to speak. If you respond to indomethacin, you may have one of these disorders.