Learn Spanish (Latin America): Me da miedo disparar.

You would change the main verb da to the conditional perfect form of dar, which is habría dado. The sentence becomes: Me habría dado miedo disparar. You would use this to make a probable guess about how you would have felt in a past situation that didn't actually happen.

I usually use tener miedo for being afraid. Why does this sentence use dar miedo?

You use tener miedo when you are the grammatical subject experiencing the fear (e.g., Tengo miedo de disparar - "I am afraid to shoot"). Dar miedo functions backward, exactly like the verb gustar. The thing causing the fear is the subject. Literally, this sentence translates to "Shooting gives me fear."

Why is the verb da conjugated in the third-person singular?

Because the grammatical subject of the sentence is the action of shooting (disparar). In Spanish, whenever an infinitive verb acts as the subject of a sentence, it takes a third-person singular verb. If a plural noun scared you instead of an action, the verb would become plural: Me dan miedo las arañas (Spiders scare me).

Since expressing fear triggers the subjunctive, why is disparar in the infinitive here?