"We like sport."
Translation:Мы любим спорт.
In American English, we do not use "sport" as a term for sports in general. We either like a sport or sports in general. So,"I like sport" is not proper in American English.
In short, yes.
I think in formal Russian pronouns are almost never dropped.
In colloqual speech, they may be dropped, but I don't think it depends on the verb form. We also drop them before past tense sometimes, and past tense doesn't make it clear. Unfortunately, I don't know any rules for dropping them, I just rely on my intuition.