Do ranged weapon feats (like Rapid Shot, Many Shot, Deadly Aim, etc.) apply to throwing darts, or do they only apply to weapons like bows, crossbows and firearms?
almost all of the feats you seem interested in specify "ranged weapons" specifically so most would work with Darts.
However one exception of the feats you listed specifically works with bows only. Manyshot says:
When making a full-attack action with a bow, your first attack fires two arrows. If the attack hits, both arrows hit. Apply precision-based damage (such as sneak attack) and critical hit damage only once for this attack. Damage bonuses from using a composite bow with a high Strength bonus apply to each arrow, as do other damage bonuses, such as a ranger's favored enemy bonus. Damage reduction and resistances apply separately to each arrow.
Emphasis mine. This feat specifies it works with bows and arrows only. If the feat is not meant to work with throwing darts it will likely specify in some way like the above.