The function of なんか is to express a negative feeling towards the word immediately preceding it. It indicates a dismissive sentiment that the preceding word is worthless or undesirable.
So in the case of カラオケに行きたくない it makes sense to place it in either of the locations you've mentioned, because both karaoke itself (カラオケなんか) and the desire to go there (行きたくなんか) are undesirable in the speaker's opinion.
However in the case of あなたを失いたくない, it only makes sense to put it in the second position あなたを失いたくなんかない. This is because while "wanting to lose you" is an undesirable notion that can be vehemently dismissed, "you" is not. あなたなんか失いたくない would mean something like "I don't want to lose someone as worthless as you", which doesn't make much sense.
For the same reason, you won't usually see this なんか attached to a non-negative construction - because it's dismissive, it's only really natural to use it when you're negating the attached word. Additionally, I don't think なんか can be attached to actual verbs, only nouns and adjectival/adverbal constructions, so it doesn't attach easily to positive constructions in a grammatical sense either.
The related なんて, which can convey essentially the same sentiment, attaches more easily to verbs and positive adjectives, but again since it's dismissive it's not commonly used without some kind of negation (eg. カラオケに行きたいなんて思わない。 or カラオケに行くなんてごめんだ。) When it is used in positive constructions, it usually expresses an implied disbelief (eg. 彼がカラオケに行くなんて… is positive on the surface, but there's an implied ありえない or 信じられない or 馬鹿げている etc.)