1. many people also make the mistake of believing that listening is a passive activity. Is that a
adjective
adverb
preposition
conjunction
2.competent delivery is simply direct, friendly, and enthusiastic. Is competent
adjective
adverb
preposition
conjunction

1. that is -------- I don't know. (If I had to go with one of the options I'd go for adverb as it describes how they are believing.)

2. Competent is an adjective.

I think that would be a conjunction because it's joining together two groups of words that could both stand on their own.