the EXIF data is written INTO the file itself. IF you had the camera setup to write that into the EXIF fields, and IF you can recover the file , then it is possible that you can recover that information from the file.
Yep, most likely if you managed to recover an unedited copy of the original image file, it would also contain the EXIF info, which will tell when the picture was taken, along with the various camera settings. You can look up the EXIF info using many image viewers, such as Irfanview. That is, of course, if you can revover the image file, which may or may not be possible depending on if that part of the storage on your hard drive or device were overwritten with other files in the mean time.