All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
A person`s ownership of his property is linked with his life. As long as he is alive, all his property, movable and immovable, is his. Based on this, your mother`s (Asker`s) property is hers because her having Alzheimer`s doesn`t affect her ownership of the property; rather it affects her capacity to have access to that property.
In this situation, it is incumbent upon you to go to court so that the judge appoints a trustworthy guardian to manage her affairs and preserve her wealth. Moreover, it isn`t permissible to spend her money but in her best interest for spending it otherwise is null and void. Also, one who does the latter is sinful since an incapacitated person, although old, is similar to a little child in the sense that it isn`t permissible to donate any portion of his/her wealth.
Al-Shirbini says, "It is the duty of the incapacitated person`s guardian to act in his/her best interest because Allah the Almighty says {What means}: " And approach not the wealth of the orphan save with that which is better, till he reach maturity. Give full measure and full weight, in justice. We task not any soul beyond its scope. And if ye give your word, do justice thereunto, even though it be (against) a kinsman; and fulfil the covenant of Allah. This He commandeth you that haply ye may remember." {Al-An`am, 152}. He also says {What means}: " Upon the world and the Hereafter. And they question thee concerning orphans. Say: To improve their lot is best. And if ye mingle your affairs with theirs, then (they are) your brothers. Allah knoweth him who spoileth from him who improveth. Had Allah willed He could have overburdened you. Allah is Mighty, Wise." {Al-Baqara, 220}. If the guardian isn`t sure whether his action is in the best interest of the incapacitated or not, then he must stop it. This same view was also stated by Sheikh Abu Mohammad and al-Mawardi. The guardian is required to invest the wealth of the child and protect it against damage. The same ruling applies to the wealth of the mad and the spendthrift." {Moghni Al-Mohtaaj, V.3:P.152}. Article (212B) of the Jordanian Personal Status Code states, "The affairs of the incapacitated person or the child shall be handled by his guardian or custodian."
In conclusion, it isn`t permissible for you (Asker) or your mother`s guardian to use her money for building a mosque or donating it on her behalf as long as she is alive because, in principle, you must be keen on preserving her wealth until Allah blesses her with recovery or she dies then her estate is divided among eligible heirs. In case your father or anyone else disposes of her wealth without the judge`s permission, then he must give it back. We (Iftaa` Department) advise all of you to fear Allah and avoid incurring His Wrath. And Allah the almighty knows best.
