Currently, Instagram does not support https://instazoom.io/ animated GIFs as profile pictures. However, you can record a short video and use it as a profile video to add some movement to your profile.
Yes, Instagram uses a square profile picture with a recommended size of 110x110 pixels. Make sure your image doesn't exceed these dimensions to ensure it displays well.
No, the change profile picture feature is only available on the mobile app. Open the Instagram app on your mobile device to customize your profile picture.