On compression yes but you can craft a jpeg image which has exactly the pixel data you need. It will fail if someone recompresses your image, but that would be unlikely
Not unlikely at all, if you're going to post it on social media
Twitter recompresses images, as do most networks.
Two or three bits per pixel, grayscale, should be quick and pretty resilient. A hundred kilobytes or more is easy to transfer.
Or, how about a QR code?
If you want absolutely terrible density, sure. Still beats 140 characters.