When baking, is it best to let eggs come to room temperature before adding then into the mix?

Absolutely. When making a recipe that involves creaming butter and sugar, the room temperature eggs will emulsify with the fat creating a better crumb. Eggs also foam (whip to full volume) better when room temperature. This includes whites. Whites always whip to better volume with room temperature, warm and are older.

And, you can place them in a bowl of warm water to speed it along.

Not only eggs but most ingredients at room temperature will help all the ingredients incorporate better

Yes. The exception is when you are using egg whites for whipping. They whip better cold.