Does it count as uploading images, since there's an image when you're updating them? I don't want to crash the site when I edit them. I mainly want to change the classification of the characters.

No, why should it? If you're not changing or adding a picture it is not.

And no it certainly wouldn't crash the site. It's supposedly because since we are technically on the old server again the new images won't be transferred for any reason if the new server is ready to rock.