Hello there, may I help you? You need to speak with your ex wife about your child and see if she has also witnessed this behaviour. Your common ground is that you both want what is best for your child. I recommend talking through your concerns with the health visitor too.
It may be that as your ex wife has an eating disorder she is witnessing her behaviour around food and mirroring them. Dont react to any problems with food, alway sit with her, make meal times sociable and fun, chat with her about what she did that day or what you are doing the next day. Make food small portions so she is not over whelmed, interesting and varied. Ignore the putting fingers in her mouth too. If she refuses to eat, finish your meal and clear away without reacting.
Later try leaving healthy snacks on the table that she can come and eat such as some fruit or vegetables and dip. Let her know it is there if she wants to come and nibble! If she does just join her and chat normally.
The laying on the floor refusing to continue an activity is fairly typical three year old behaviour, as you says she snaps out of it, just ignore it, continue the activity she was doing by yourself talking aloud about what you are doing, if she realises she is not getting a reaction she will quickly want to join you and play.
The main concern here is the time spent with her mother and the negative impact her illness and depression may be having upon her. Let your ex wife know that you want to help as much as you can with your daughter and aim to work with her. If she is suffering from depression you dont want her to feel that you are concerned about her parenting. Keep the lines of communication open and positive as possible.
I hope this has helped, do let me know if i can assist you further.
Thanks for your help