Can Botox under your eye make your eye look more open?
In some people Botox can be injected beside the eye to reduce the pull of the muscle there on the brow allowing the brow to elevate slightly and, thus, make the eye look more open. Placing it below the eye will not do so. It will also not work if gravity is more of a force than the lateral orbicularis muscle. If the eye looks "half closed" this may be the case and botox may not work. If so, you need a brow lift. Make sure your surgeon has significant experience in use of Botox and in brow lifting.
Around the eyes we can use the Toxin to raise the brows and for the craw feet. If we raise the brows the eyes will open a little more for sure, but if you are a candidate for a Blepharoplasty, it will just be solved with surgery. Try to choose an experienced ASAPS member Plastic Surgeon.
It is difficult for us to answer without photographs and examination, but in my experience Botox is not commonly used under the eyes. You mentioned droopy eyelid appearance. This is often the result of loose, redundant upper eyelid skin - and is treated surgically with a cosmetic blepharoplasty. Occasional patients have droop caused by stretched eyelid lifting muscles, and a different surgery is done to correct that. Your surgeon should be able to evaluate this and make recommendations.
Botox, and other botulinum products, temporarily weaken muscles in specific areas. It is useful for decreasing crow's feet wrinkles beside the eyes. It can give a subtle amount of lifting to the inner or outer eyebrow. It's use can improve the appearance of the area surrounding the eyes. It can compliment the benefits of eyelid surgery.