Learn English – Double negation “does not any”

To answer your first explicit question, I would say they are double negatives:

A double negative occurs when two forms of negation are used in the same clause.

To answer the second question, I would say the use of litotes is perfectly acceptable.

Litotes is a form of understatement, always deliberate and with the intention of emphasis. However, the interpretation of negation may depend on context, including cultural context. In speech, it may also depend on intonation and emphasis; for example, the phrase "not bad" can be said in such a way as to mean anything from "mediocre" to "excellent."

The respective Wikipedia articles (linked to and excerpted above) give a lot of good information. I would like to emphasize the potential ambiguity in litotes, in that the intensity of the double-negative-as-positive ranges from "mildly positive" to "resoundingly positive".

Finally, see this other EL&U question covering the specific example of not uncommon.

Your option #1 is much closer to the intended meaning, but you haven't quite construed it correctly. The two negatives do indeed cancel each other out. But when you cancel out the word never the result is always, so the plain un-negated sentence would be something like

He always dreams of thinking of himself as sexually adequate.

However, this still misrepresents the meaning. This is because the idiom to never dream of carries a negative connotation, and when you un-negate the sentence you should take out that idiom entirely. So the plain meaning of the sentence is more like the following:

He always thinks of himself as sexually adequate.

Or, to put it more strongly:

He is confident in his sexual adequacy.