Android - Can I make a full backup of my phone with Odin and without root?

Odin is Samsung-only, you're right. (And as far as I know it isn't for all Samsung phones either, though I could be wrong on that.) Since Heimdall is, as far as I know, just an unofficial port of Odin I would assume it's the same, but I've never used it.

Odin3 has a "Dump" function that I had never used. It seems to be disabled in v1.8x and v1.7, but not in v1.52.

I just tried it in v1.52 and it didn't work. You hit the Dump checkmark under Option, select an output file via the Dump button under File [Dump], then hit Start. Odin will say "OneNAND Dump Start" or similar as it backs up the device. It only backed up 1 KB of something before halting, though. Odin just sat there without giving an error message.

You might have different results with a different device (I have a Vibrant). I'm also wondering whether my Ext4 lagfix might be affecting this. Perhaps later I'll flash to stock 2.1 and try again. Edit: No change on stock 2.1 JFD.