During my five-year relationship with a woman, I’ve lied to her, lied about her, cheated on her and was partially responsible for her losing her job. She has a child, 6, who looks to me as a father figure and is the main reason the two of us are still together.     
      Because of my actions and my past actions, we fight often and the fights have grown in both intensity and frequency. It’s turning her once loving, peaceful home into a battlefield.     
      We talk about remaining friends, or it’ll have a detrimental long-lasting effect on her child. Am I crazy to think that we can work something out?
      - Confused

Since you’re admittedly the perpetrator here, creating deceit, distrust and chaos, a lot of the “working out” of things is up to you.     
      She should’ve thrown you out long ago, so I have to assume that the negative effects the child must already feel from the fights, puts you both at blame on that score.
      Staying together will only restore peace IF you’re capable of not lying, cheating, etc. However, if you feel this woman is not for you, and that’s what causes your shabby behaviour, do the child and the mother a favour and leave.     
      Currently, the relationship is unhealthy for everyone involved, especially the child. You can still visit her and keep contact.

I feel very used by my sister. She only calls me when she needs something. If I call her, I either leave a message (which she doesn’t return), or if I do speak to her, she says she’ll call back as she’s busy. Or she spends part of the time on the phone talking to her kids or husband.     
      When I mention this she says, “ Sorry, I'm busy with the kids,” or something. She has time to go out with her neighbour friends, and other girlfriends.    
      I’m allocated a once-weekly 5-7 minute phone time while she’s driving to work, with the conversation muffled via a hands-free device.              
      I’m tired of feeling used and feel sad, as I don’t have any close girlfriends I can turn to. My feeling is to distance myself from her and give her the same treatment back – and not return her calls if she does phone me.      
      Any attempt at a serious conversation about this, she gets mad and puts me down with, “you’re too sensitive.”
      - Hurt Sibling

An objective read cannot miss certain realities: Your sister IS busy, you’re relying on her for more than she’s capable of giving and you’re by nature a sensitive person who takes this personally, though she likely doesn’t mean it.
      The likely reason she has some time for friends is that they’re likely doing similar things. Understand the time constraints without feeling hurt and do not have a history of siblings that affects your relationship from the dynamics of the past.    
      Meanwhile, she DOES try to devote some time to you once a week and that shows she cares. The muffled sound is not on purpose, so stop looking for put-downs.     
      Turning this around by not calling her will achieve nothing, but a longer, deeper rift, for no good reason. Call her less and save your news for that weekly call; when she “wants something,” only do what you can in response.
      Try to arrange a visit to see her kids every now and then, so you and they can enjoy the benefit of an extended family and the fun of being an aunt.

What do you think of a man wanting his joint custody time spent with three youngsters (sisters), when only one child (age 2) is his own? The other two girls both have different fathers.     
      The mother never married. The oldest child, 6, has a father who also has joint custody; but this man insists that all three sisters are together.    
      The mother feels any conflict of time between the two fathers must be worked out between them.
      - Odd Demand?

The man is pushing for control, which is why the mother is resisting. But he may feel his toddler is more comfortable when her sisters are around, and/or that it’s easier for him to provide an activity with other kids along.     
      Since it’s important that the child and father get together, it shouldn’t be that hard for the mother to try to comply, or to assist the two fathers to co-ordinate their time.