All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
First of all, residing in a non-Muslim country has many disadvantages on the Muslim in his religion as well as his worldly life, and this is something that is very clear for every person who has a sound mind. Therefore, migrating from it to a Muslim country is something Islamically required.
This migration may be permissible, desirable or obligatory according to different circumstances (of the person) as clarified by the jurists . For details, please refer to Fatwa 86405.
In your case, the least that can be said is that migration in desirable for you.
On the other hand, there is a difference of opinion among the jurists if the mother wanted to move to another country whether or not she loses the right of custody of her children. However, if we presume that you lose the right of custody, then if the father is dissolute or non-Muslim, or that the environment in which he resides is corrupt while there is no fosterer for your daughter, then it appears that it is permissible for you to travel with her as the benefit of the fostered child comes in priority.
As regards the daughter of your husband from his ex-wife, then it is not permissible for your husband to leave her with her mother as she will influence her and nurture her on corrupt beliefs. Her mother has no right in her custody as long as she is non-Muslim. According to the most preponderant view of the scholars among the conditions of custody is that the parent must be a Muslim.
Therefore, your husband is obliged to make every effort to take her from her mother. If he can do so, then this is what is required, otherwise he can migrate and leave her with her mother. In any case, he should be keen on keeping ties with her and asking about her especially when she grows up hoping to gain her affection, and so, he may influence her.
Allaah Knows best.