In these situations, I turn into Mr Rogers.
Seriously. If you watch his show with a critical eye toward how he speaks to the child viewer, you'll see he's very authoritative.
"Let's do this now."
"Let's go to X... come along now, here we go."
"Now we're going to......."
He's extremely calm and matter of fact. I have a preschool teacher friend like this, and it even works with adults. I think of it as the way a tour guide might talk, very directly providing guidance so that no one gets lost.
This is what I do when I have children other than mine over. I give matter of fact "this is what we are doing now" guidance. When it comes to picking up toys for cleanup, I specifically give each child their own job. "Kiddo, I need you to put the cars in the basket. Friend, you are in charge of the Legos."
When it comes to nagging, I do not put up with it.
First request: 'Can I have the cookie?"
Answer: "We're not having cookies right now/today. THIS is what we're having."
Second request/complaint: "I want a cookie."
Answer: 'Oh, yes,you do. We're not having that today. X or Y is what I have for you.
Third time: I really want a cookie!
Answer: Are you asking because you think I am going to change my mind?
(this is usually a showstopper. It pins it right on them.)
Child: yes!
Answer: Well, I'm not changing my mind--you may have THIS. I'm all done talking about it. I'll be happy to talk about something else when you're ready...
I say all of this calmly, putting the problem back on the child. Do they accept THIS (whatever you are offering) or do they pout and sulk? If I'm actively ignoring, I'm pleasantly talking about something else or I'm being quiet and listening for a change of topic, and then I will join the conversation pleasantly, pretending nothing ever happened.
I have one child who always gets evasive and doesn't want to leave when their parent comes. I often position my body between that child and my son's room. "Oh, we're going to get shoes on now. Let's go get them." I don't offer a lot of wiggle room.
How I build trust with the child is to notice what they are doing when they are cooperating, either through a verbal recognition and interest ("I see your playdough creation is huge. Tell me about that." or "Thanks for keeping your food at the table, all of you.") and through smiling, non-verbal approval when I see them happily busy, being helpful, or being friendly with their words and actions. I do praise a lot when I see or hear appropriate behavior. "I liked the friendly way you asked me for that." or "Thanks for clearing your place." I ask them about their interests, or to expand on story they are telling.
As for your last question: Do say simply "let's try it out for a couple weeks, just to make sure this works for everyone." Make sure you are both keeping track of hours, too, so that everyone feels they're getting a good deal on this trade. I don't think it needs to be a friendship buster. Do let your friend know that you run a pretty tight ship and that your home is pretty structured, and unless she balks, I'd say, go for it. When/if you feel the little girl is needing some discipline or isn't getting along, I'd set her at the table with something to do. "I need you here for now..." If she fusses, offer sitting and having a quiet time with some books for a while. Give her options: "It's okay to do X, and we don't do Y at our house." She may need to be told what is appropriate since her mother is not asserting this sort of teaching at home.
And lastly, if you are really dreading this-- just don't do it.