Welcome! It sounds as if you are pretty frustrated with your daughter laughing when you are trying to correct her behavior. Many times children her age will laugh if they feel anxious. When she has had time to calm down some, she probably cries because she is upset because she failed to meet your expectations.
Another way to approach behavior challenges in children is to focus on what they are doing right. When you "catch her being good" and praise her, she is getting you attention. Make sure you also remember to tell her what to do....when children hear no without direction on what to do, they are likely to just get into more mischief.
As far as her not doing this at her father's house, are you sure this account is accurate? Children sometimes like to play one parent against the other, especially if they perceive the each parent has different expectations. If you have not already, you might consider discussing the situation with her father. It might be that she is doing the same thing at his house.
I hope this information is helpful to you. Please let me know if you need any clarifications. If not please remember to accept my answer.