Yes, if you injected botox under the corner of the eyes rather than in the crow's feet, it will cause puffiness and swelling under eyes. In fact, eye swelling, puffy eyes and even burning eyes are the side effects of take botox thought they can enhance your skin. So, be careful if you plan to injected botox. Good luck.
Yes, generally speaking, the swelling eyes after the Botox injection are very normal for many patients. So you should know the cause of this situation. There are mainly two reasons which can cause swelling eyes from Botox, such as Edema from injections and Paralysis of Orbicularis Oris Muscle. For your situation, you can just wait for several days, they can go away by themselves. And if the situation still exists, maybe you should go and see an eye doctor as soon as possible.
A little amount of botox fluid suspend at the eyelids after botox injection, which makes your eyes looks swollen. This is one of the common complains of the treatment. In most cases, you can get rid of it once the fluid has been assimilated. It should dissipate several days after the treatment. If the swelling lasts longer, you need to inform your doctor.
Cold compression using an ice pack might improve the situation. Actually, my recommendation is doing nothing. Leave it. For a few days, your eyes will heal.
Yeah, usually the first time of botox injection may cause the eyes get swelling. However, it will get disappeared for a while. You could wait and see. If there is no releasing symptom, you'd better go to see the doctor and have a check on the eyes, avoiding to get the infection.