I somewhat agree, though I think "self-confidence" is slightly different from "confidence" in that it generally means something like "confidence in one's own value or worth" or "confidence exhibited in one's general approach to life"... it's used specifically to refer to a character trait, whereas 'confidence' can refer to a disposition specific to a particular instance. One might be unreservedly confident of one's ability to pass a test, but that doesn't necessarily imply that one has a lot of self-confidence.
My guess is that "self-confidence" formed in a way similar to how "unravel" came to be a synonym of the older word "ravel"... because ravelling is a sort of "undoing", "un" seems a natural prefix, and eventually it became "correct" -- to the extent that one rarely hears the word "ravel" in contemporary usage.
"Self-conceited" is a phrase I'm not familiar with; at first blush, it sounds redundant and just wrong. I imagine that it results from a muddled conflation of 'conceited' with such terms as 'self-important', 'self-aggrandizing', etc. Offhand, I can't think of any situation where I would prefer 'self-conceited' to 'conceited'.