I think you definitely have to work on getting her more used to other people. Whether that is a church nursery, your parents, some friends of yours, whomever you trust. Tell her that you love her and that you will be back later. Give her a kiss or hug and then walk away. It is hard to let them kick and scream like that but they eventually get used to the idea that they can have fun with other people and that you WILL be back.
As far as her dad goes, I would say that it depends on him and court orders. As long as you know that she is safe there, she should have to go. If you suspect that she is being abused, etc. there and that is why she is fighting it, then that should be investigated more.
Well for one she's 5 and you have children, when you have kids the thing you give up to keep up with them is your Life. She seems to be very attached to you and if shes only seeing her dad 2x at the most a month she might not be comfortable staying or going with him , if she feels hes a stranger. She's still very young , there could be an issue with her and him , that you might not know, you should try asking her why she doesn't want to go. Forcing her could make her angry or hurt on the inside .. if she really does not want to go , don't force her. Sharing a room could be why shes so attached she needs to able to be on her own. Be independent she needs to learn there is a life away from mommy and that mommy need to work to take care of her and that moms need fun to. My little brother is 5years old and is very attached to me & my mother, and when we leave he throws fits because he's always included when we go out and when we leave him he says nobody wants to take me because they don't like me. If shes throwing fits , don't feed into them. Temper tantrums at 5years old should be shut down , just remind her you will be back soon and that shes to young to go out.
I think that when parents seperate, it can be very hard on kids. They feel vulnerable and less secure. And then when they go back and forth, it sets up a feeling of instability. No way around that as it is the nature of divorce and just what happens. But, we can as parents keep in mind how it must all feel to our kids. Some handle it better than others. Some will act out.
What I'd try is distraction. I'd have a plan that will be exciting to her for when you go out with the girls. A special movie she has been wanting to watch, Grandma playing her favorite game with her and getting it started as you head out the door, a fun craft (even painting a rock is super fun to a 5 year old!) to do as you leave, etc. The art of distraction is one to never be underestimated when it comes to a 5 year old. By understanding that she has lots of feelings and doesn't quite know what to do with them will help you deal with it better. So put yourself in her place and think like she would.
When you are home, I'd do lots of reassuring with her. Making her feel safe is important. Letting her know that you aren't going to leave and live elsewhere like dad is important. Letting her know that dad still loves her and will be there for her, etc. All that kind of positive communication.
Hopefully she will outgrow this quickly. good luck