Videos via MMS will be compressed because MMS was never meant to send high-quality video (and it's an old standard). And no, iMessage should not be compared to MMS because it's not the same protocol. iMessage should be compared to IM services like Whatsapp, Skype, or even the new Advanced Messaging standard (but, at least to my knowledge, only a handful of TMo devices support that). As for e-mail, there might be a 'switch' in your app to automatically compress attachments to make them for suitable for this (most e-mail services have caps on attachments).
That being said, I just uploaded a video from a Note 5 to FB and it is showing in HD. So not sure why yours isn't. There is a bit of visible compression artifacts introduced by FB but nothing major.