Where is a video stored after it is deleted from my iPhone? If I take a 60 second video and send it to someone via SMS/iMessage and then delete the video from my Photo Library, where is the data stored?

Is it still stored on my iCloud, the iCloud of the person who received the message, both, or is it possible that it is stored somewhere else?
The video is then stored in your Messages application as opposed to your photo library. If you have iCloud enabled for messages it is stored in iCloud as well for both people, since you both have the message.

To see photos stored in Messages, go to [Settings] -> General -> iPhone Storage -> Messages, there you will find a section for Photos and Videos: