It depends on the dialect. In standard Dutch we always pronounce the "n" at the end of plurals.
I don't have the impression that this is standard Dutch. Maybe it used to be, but I mostly hear people not pronouncing the final n. Pronouncing it sounds a bit overly formal to me, except when you're from the eastern part of The Netherlands.
Can I say "Ik ben op zoek naar een asteroide" ?
In some exercises it's "zoeken naar". In others it's "zoeken". What makes the difference? How do we know if we need to add "naar" or not?
Given the Dutch tendency to not pronounce n's at the end of plurals, is their any difference in pronounciation between asteroïde and asteroïden?
I think you'll only hear the final n when the next word begins with a vowel. Or when the speaker is, for instance, from the eastern part of The Netherlands.