I'm on doxycycline for hidradenitis suppurativa, taken half the doses out of 30. Having severe cramps and bright blood in stool. Can I stop taking it?

Probably. Doxycycline has alot of GI side effects. Because you are using it for a skin condition you likely need to stop it to alleviate the cramps. Not sure about the blood in the stool. This may or may not be related.
The blood could be from a hemorrhoid and not related to the doxycycline. Do you have any pain in the rectum.? Was the blood mixed with the BM?
Need to see a Dr. for check up, and anoscope.