All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.
First of all, we ask Allah to make the situation in your family as best as possible and to keep the plots and whispers of the devil away from you.
We also ask Him to foil the plots of the devil in trying to separate between the loved ones and disunite the family, as the devil is the enemy whom Allah warned us against. Allah says (what means): {Indeed, Satan is an enemy to you; so take him as an enemy. He only invites his party to be among the companions of the Blaze.} [Quran 35:6]
Jaabir, may Allah be pleased with him, narrated that the Prophet said, “Indeed, Iblees (Satan) places his throne upon water; he then sends detachments [for creating dissension]; the nearer to him in rank are those who are most notorious in creating dissension. One of them comes and says, ‘I did such and such.’ And he says, ‘You have done nothing.’ Then one amongst them comes and says, ‘I did not spare so-and-so until I sowed the seed of discord between him and his wife.’ Satan goes near him and says, ‘You have done well.’” [Muslim]
The devil – who is the enemy of Allah – may take advantage of the love of the woman for her husband and her jealousy regarding him and make her suspect him. In principle, a Muslim is innocent from any accusations until proven otherwise. Allah says (what means): {O you who have believed, avoid much [negative] assumption. Indeed, some assumption is sin.} [Quran 49:12] Please, refer to fatwa 89109.
What you mentioned does not prove that your husband has an affectionate relationship with your sister, and we cannot be sure that your husband's actions suggest an affectionate relationship with your sister, and we cannot totally deny it either.
However, if we were to presume that this was proven, then it would first be necessary to remove the means that lead to this evil, as this matter usually occurs due to violating the teachings of Islam and due to the husband being too casual in dealing with the non-Mahram (marriageable) women who visit his home, especially the female relatives of his wife. He must deal with them according to Islamic conditions.
Therefore, you should advise your husband and your sister in a gentle and soft manner and remind them of Allah. Also, you should supplicate for both of them with goodness, as Allah has ordered us to supplicate Him, and He has promised to answer our supplication. Allah says (what means): {And your Lord says, “Call upon Me; I will respond to you.” Indeed, those who disdain My worship will enter Hell [rendered] contemptible.} [Quran 40:60]
If your advice is of no avail to your husband and you dislike to remain with him and are afraid that you will not be able to fulfill his right upon you because of this, then in that case it becomes permissible for you to ask him for a Khul’, as the wife of Thaabit ibn Qays, may Allah be pleased with both of them, did with him.
However, be aware that the benefit is not always found in divorce. Being patient and trying to reconcile may be better, especially if you have children. For more benefit, please refer to fatwa 85044.
With regard to spending, the husband is not obliged to give his wife money for the household expenses if he does the required spending himself. In this case, his wife has no right to ask him to give it to her. On the other hand, if he does not spend on the house and refuses to do so, then his wife has the right to take from his money even without his knowledge, as confirmed in the authentic Sunnah. She is also permitted to ask for divorce for this reason.
Allah knows best.