A.C.
I struggle with this with my son, too.
First, don't ask her to do things. Asking implies that she has the right to say no. If she needs to do things, just tell her "Put your shoes away" or "Shoes go in the shoebox." Some kids really view you saying "Please do X" as a request, which they then think it's okay to say No to. So I suggest dropping the please unless it's a really question and you don't care if she answers No.
For the cheese example, if my son whines, I tell that I can't understand him when he uses that voice and to ask in a nice way. Once he asks "can I have a piece of cheese" (or better yet, please, can I have a piece of cheese) in a normal voice, I reward the behavior. If it's do-able and against something I already said.
I have to really remind myself to try other ways to motivate. If it's time to get in the car, for example, and I think it's going to be a struggle, instead of saying "Time to get in the car" I say "Are you going to hop like a frog or dance like a ballerina to the car?" or "What song should we sing while we walk to the car." Or "How fast can you run to the car" or "I'll race you to the car!" If you can make it a game, then she's likely to opt in.
One thing that works well (though I try to use it sparingly) is reverse psychology. "I bet you can't do X".
Sort of related to the above is "Can you use your strong muscles to ...?" or "Can you use your loud voice to...?" 3yos like to think they are getting bigger and stronger/faster/etc because they want to be "big kids" so you can use that to your advantage.
I'm sure there are other things that work that I can't think of right now...