I'm having some trouble with my 2005 Hyundai XG350L's gas mileage, any suggestions?

The horn probably cannot be repaired. For proper function, I would replace the horn.

I cannot provide accurate data regarding the vehicle appearance/body as I do not have access to the technical manuals. For best results, consult a reputable body shop.

I fixed the horn, the housing was down in between my radiator and bumper so I just superglue it back on and it works fine.

Would driving in sports mode and shifting as soon as possible be less beneficial to gas mileage than driving in automatic?

The difference would be negligible.