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 is His slave and Messenger.
Firstly, if your husband divorced his first wife and she became irrevocably divorced from him, then she has become non-mahram (alien and stranger) for him. Therefore, it is not permissible for him to live with her in the same house unless each one of them lives in a part of the house that has separate utilities. For more benefit, please refer to fatwa 38704.
Nonetheless, there is nothing wrong with him contacting her for a need, such as for the benefit of the children and so on, provided they both abide by the Islamic limits while doing so and avoid reasons of temptation.
Secondly: If this house belongs to your husband, then there is nothing wrong with him letting his ex-wife and his children live in it. The father is obliged to provide accommodation for his children who are in custody, and some scholars hold the view that it is an obligation on him to also provide accommodation for the mother who has custody of them if she has no house.
Ibn ‘Aabideen said, “The sound opinion is that the person who is obliged to spend on the fostered children is also obliged to pay for the accommodation fees, because the house is part of maintenance, but this is in case she [the mother] has no residence; however, if she does have a house where she can foster her children and they live with her in that house, then he is not obliged to pay for the accommodation fees as he does not need to do so.”
This is when he is obliged to do so, but if he wishes to let his ex-wife live in that house in any case, then he has the right to do so, as she is the mother of his children; being generous to her is being kind to the mother of his children. So you should endeavor to repel any whispers that may come to your mind because the devil may cause sorrow for a Muslim in order to make his life miserable. Allaah says (what means): {Private conversation is only from Satan that he may grieve those who have believed, but he will not harm them at all except by permission of Allaah. And upon Allaah let the believers rely.} [Quran 58:10]
For more benefit, please refer to fatwa 163965.
Thirdly: your husband is obliged to spend on you and on his children, so you are not obliged to spend anything except out of your own proper desire. If you help your husband, you will be rewarded by Allaah, and this may be a reason for gaining his love and making your marriage stronger.
Hence, you may come to terms in a gentle manner with your husband in every matter relating to the family, and you may ask for your rights, but you should not get involved in that which does not concern you; among which is what he spends on his ex-wife and his providing her with housing. Also, you should avoid what may provoke him, and this is better for you, Allaah willing.
On the other hand, you did well by urging your husband to see his children and by you purchasing gifts for them; this is indeed proof of your intelligence and wisdom and good manners, and you will find the reward for this in this worldly life and the Hereafter.
If his ex-wife insulted you and incited the children against their father, then she is very wrong, and she has committed a grave mistake against her children by raising them on being disobedient and undutiful to their father. Allaah says (what means): {And cooperate in righteousness and piety, but do not cooperate in sin and aggression. And fear Allaah; indeed, Allaah is severe in penalty.} [Quran 5:2]
Allaah knows best.