Not sure if this is worth pointing out, but you can't really send a picture via a text message ;-)
An alternative (no direct translation) would be to say: "Send bildet enten som (en) melding eller på e-post". Melding covers both SMS and MMS and it would be understood that you talk about a phone message.
Indeed you can! At least on Verizon...
Hmm, what about the word "mediemelding" instead of "tekstmelding"? Would that be right?
"Multimediemelding" is usually shortened to "MMS", and may be used for any "text message" containing images or video.
I've never heard just "mediemelding" used to mean that, and would assume it referred to some sort of message for the media.