No, I do not think you should accommodate as hoc visiting requests. It is not fair on DD and it is not fair on you. What is best is consistency - that is to say, it is best for DD to know when her dad is coming and for you (and she) be able to plan her day to day life aside from this. I found it really important to ensure routine - both with seeing dad but also routine time for extracurricular activities, seeing friends, doing exercise, whatever, so that life goes on regardless of the emotions, but also you build up your social network aside from your ex. Routine and consistency.
My DD’s dad works shifts so I do understand the problems, but he sends his rota in advance, and DD decides when she wants to go fitting in her own commitments. She is a bit older and when she was your DD’s age, we used to just agree the days when he sent over the rota. So it was easier that way to know at least a month at a time. So even a routine of him sending his rota when he has it, and then you agreeing on days which go in the calendar for the next month will be a step forward. How much contact and how often depends on your DD’s wishes and her other commitments but it needs to be manageable within her day to day life.
I personally would make the appointment for March with the counsellor. You will need it then as you are just in early days and it is really raw still. In the meantime keep a journal (I found this very helpful) and think about things you would enjoy in the time you have in the evenings - reading, art, knitting, writing, what makes you tick? I agree with seeking support from your friends.
Not sure if that is any help, but you will get there.