Here is another area without "rules". I am a divorced mother, too, and have several friends who are or have been divorced/single. Sometimes, meeting right away works - for instance, one of my friends met her current husband at a gathering of friends and their families, so her daughters met "the guy" the same evening she did and they started having family outings from the very beginning. In other situations, months may need to pass before a meeting happens.
I absolutely tell a man that I have a daughter before we even "date", as she is an important part of who I am, but I've found it can take several months to really know a man, and know if he's someone I'm going to be with longer term. Until this, I don't see a reason to introduce him to my daughter (she's 8 now). She's only met one "boyfriend" (current) and only briefly met him twice.
Where my situation is different from your is that her father is an active co-parent and she's with him 1/2 time. This gives me free nights to date without this involving her at all. Also, I have no intention of marrying again, having been a stepparent for 24 years and a parent for 8, I'm not thinking of "blending a family" again. So dating is for my enjoyment, not for deciding on a husband/stepfather.
When is it appropriate to "sleep over" is also very situational, in my opinion. In my 20s, I slept over often and then moved in with the man who eventually became my husband, and his two young children. We believed from the befinning it was a long-term relationship (24+ years!). Now in my late 40s, I have not had and am unlikely to have a man sleep over when my daughter is home. Partly because I have plenty of free nights for this, and partly because I have a greater understanding of the complexity of emotions that children have about their parents' relationships.
I do believe modeling a healthy relationship is important for kids. I am doing my best to do this with her father. Even though divorced, we are both intent on showing her a respectful, caring relationship.