I have a huge gap at the outer corners of my eyes and they are very droopy. (Photo)

You have lower eyelid ectropion and retraction. There are 2 ways to address this: A) ectropion surgery with skin graft; B) internal eyelid lift (eyelid retraction surgery).  See following link. See an experienced oculoplastic surgeon. Skype consult available.

When the lower eyelid is not up against the eyeball, the tears are not efficient. If the tears are not efficient, the eye can dry out which causes the redness and soreness. A weak lower eyelid is not uncommon temporarily after lower blepharoplasty, but to be born with it is unusual. A thorough examination is required, but a lid shortening procedure would bring your lower lid up against your eyeball and help to eliminate your symptoms. The lateral lid would be more closed, but would not significantly change the shape of your eye. A specialist in eyelid surgery will be able to advise you of your options. John Standefer MD.