"If the fidyah becomes obligatory upon a pregnant or nursing woman because she broke her fast out of fear for her child, is it obligatory from her own wealth, or from her husband's wealth?"
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
The fidyah (ransom/compensation payment for breaking the fast) is obligatory from the wife's own wealth, and is not obligatory upon the husband unless he voluntarily chooses to pay it on her behalf.
It is stated in Mughnī al-Muḥtāj (Vol.2/P.174): "If the two [the pregnant woman and the nursing mother] fear for the child alone — such as a pregnant woman fearing miscarriage, or a nursing mother fearing that her milk supply would decrease, thereby endangering the child — then both the qaḍā' (making up the missed fast) and the fidyah become obligatory upon them, to be paid from their own wealth." And Allah, the Exalted, knows best.
What is the ruling on a person in a state of major impurity (junub) walking or eating before performing the ritual bath?
It is from the Sunnah for a Muslim to hasten to perform the ritual bath for major impurity. However, if he delays it, he does not sin provided he does not miss the prayer. It is permissible for the junub to walk, eat, drink, and sleep. It is recommended for him to perform ablution before these actions to reduce the state of impurity, although the best is for him to perform the ritual bath. And Allah the Almighty knows best.
What is the ruling of Islamic Law on supplicating with other than the transmitted (Ma'thur) supplications in prayer?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
It is permissible to supplicate (make du'a) during the prayer for anything from the affairs of religion or worldly life, and the prayer is not invalidated by doing so.
Shaykh al-Islam, the Imam an-Nawawi, may Allah have mercy on him, says: "Our madhhab [i.e., the Shafi'i madhhab] holds that it is permissible for a person to supplicate during it [the prayer] with anything that is permissible to supplicate with outside the prayer, from the affairs of religion or worldly life. He may say: 'O Allah, grant me wholesome sustenance, offspring, a house, and a beautiful young wife' — describing her; or: 'O Allah, free so-and-so from prison,' or 'destroy so-and-so,' and other such things — and none of this invalidates his prayer according to us. And this is also the view of Malik, ath-Thawri, Abu Thawr, and Ishaq." [Al-Majmu', Vol.3/P.454]. And Allah, the Exalted, knows best.
What is the ruling on using eye drops while fasting?
Eye drops do not break the fast, even if the taste reaches the throat, because the eye is not an open passage to the body cavity.