Handle Animal covers ALL circumstances of getting the animal to do something on your timetable. Even if you had speak with animals you need Handle Animal.
Not true.The spell itself says:
You can ask questions of and receive answers from animals, but the spell doesn't make them any more friendly than normal. Wary and cunning animals are likely to be terse and evasive, while the more stupid ones make inane comments. If an animal is friendly toward you, it may do some favor or service for you.
Giving a mouse a big wheel of cheese, and offering more if he goes into a building and then tells you what there's inside, can make him friendly to you, even if you are not an expert mouse tamer. That's the adventage of communicating with him.
It would work the same, if it's the mouse the one who casts Speak with Humans. Or knows a language. Then, of course, he might not want to do it, at all, if he's scared of the cat that lives in the house, or happens to be a mouse that doesn't like cheese, or whatever. Just like any other NPC: you could pay a vagabond child to go to an alley and tell you what there's there, but he might, or mmight not, choose to go, depending on the payment, how affraid he is about that alley, or his actual schedule.