I know a separated couple who have a four year old daughter. Their custody schedule was 50/50 until recently. They didn't do phone calls because it would disrupt the girl at that parents house so they would text a picture when the other parent asked. I always thought it was weird too but they still do it.

BM used to ask DH for pictures and he always felt like she only asked as proof like she would text asking DH to have SSthen2 call her before bed but he would already be sleeping so she would ask for a picture. It was aggravating and he didn't always do it but sometimes he did. But BM also called often so she was already disrupting the household anyway. I think your wife could ask for maybe one picture if it's a weekend visit and maybe one a call a week for week long visits. But if the girl is only gone from mom two days then I think she's hovering. The same goes for the dad. I can see a mid-week call and maybe a picture on the weekends he doesn't get to see her. But not an everyday thing.

She isn't having a relationship with him aside from him being the parent of her child, at least as far as the examples you have provided would show. She is concerned for the well-being of her daughter, who is being sent off to visit with a father she can't know well, and who has a criminal past.

You do realize that dad is going to be part of this kids life, right? Until the child is grown, there will be contact between BM and BD. It will likely decrease as the child gets older, but parents have to communicate about their child. If you can't handle that, then this relationship isn't going to work for you.

I think that's what being in a stepfamily is, it is two families. My DH doesn't like to look at it that way, but that is the way I see it. There is DH and BM trying to do the best for SD as possible with all their dysfunction. Then there is DH and me, raising our kids. So, in answer to your question, yes they are now a separated family. I think it's normal to be jealous, just don't act on it. My DH and his ex would talk to SD everyday at this age, and lots of pictures were sent back and forth, because she was little and cute and still doing everything for the first time. A few years later and she was bigger and not so cute, and didn't require so much attention, so it slowed down to weekly calls, and monthly picture sending, now she's a teen with a cellphone and everything is texted through SD.

I think it totally depends on how the daughter is and how long these visits are and how much time passes in between these visits.
You mention that they are week long visits, once a month, so thats 3 weeks with BM then a week with BD. In which scenario I think its perfectly normal/acceptable for BM to want some contact, be it a call or photo during that time. So long as BM is not blowing up his phone on a daily basis and disrupting his time with his daughter, then I see no harm. How often does BD contact when daughter is with BM?

I do think its important for both parents to steadily reduce contact whilst on the other parents time, particularly if the parents arent very good at communication or high conflict.

If BM is currently having contact daily, maybe try to reduce that to alternate days (first day, middle day and last day) but if both BM and BD are happy with the current arrangement then its best to leave well alone.
Ultiamtely our jobs as steps is to support our partner, whilst they do their job as a parent.

I will say...as a woman married to an NCP these calls / texts / chats / afternoon drop-bys from the BM were annoying AF. Of course, the skids were older when DH and I got together (10,12,14) and he's always been a highly capable and involved father (no prison record), and we only had them EOWE. So, for us, it felt like BM was trying to invade that and take it away...and there was really no need.

In your situation, I am guessing that your fiancé feels nervous about leaving her four-year-old with this relative stranger...especially for an entire week at a time. I bet that as she becomes more trusting and comfortable with the situation, the calls will decrease. Unless she's a nervous, crazy, controlling BM like the one we've got...