My understanding in my neck of the woods is that "Mind your Ps and Qs" is equivalent to "Mind your manners." Being on your Ps and Qs would be to make sure you use good posture, use clean language, and act hospitable to the room. For instance, you would mind your Ps and Qs at weddings, funerals, and business events. One internet source says P's and Q's comes from making sure to say Please and Thank "Q."

My understanding of Cross your t's and dot your i's is to double check your work before submitting it. For example, if you were going to send a business proposal or a resume for a job, etc. You would want to cross your t's and dot your i's. In addition to making sure spelling is correct, you would also want to double check your grammar and make sure you say exactly what you meant to say. In other words, you wouldn't rush it. You would double and triple check it to make sure everything is as it should be.