I made friends with my husband's ex-wife and that way had no troubles with who was "mothering" when. My husband has 3 sons and I played the support role, giving he and his sons as much space as I could to bond with each other (and making sure they felt welcome in my husband's and my home). If his ex-wife is at an affair, she is the mother and grandmother and I automatically put myself into support mode. It's not a competition!

If I were in your place I'd pretend I were a really good friend of the family and (1) do whatever would help your husband with his obligations/wishes toward his daughter (2) do what you think would be best for the daughter if there's an absense of either of her parents. There's plenty of time/room for a relationship with her in there.

Don't worry about what his ex-wife says about you, you know the "truth" so it's like some "stranger" saying your mother wears combat boots :-) It's not only not true, it's not relevant/is almost nonsensical. My husband's ex-wife has said we're "best friends" but my opinion of her is not quite that good :-) But I can be friendly and helpful and understand her "limits" and help her without hurting her by making fun of her or telling her her faults, etc. I'm a good person and know who I am and how I want to behave and be and that's all that is important. I love my husband and he loved her and they made 3 wonderful children; I can support that.