Is the marriage, which lacks a valid legal contract, a guardian, and a court registration, valid?
It is incumbent that a valid marriage contract be concluded in the presence of a guardian and two trustful witnesses, and it should be registered in the court to protect the rights of the wife. Actually, a valid marriage contract is what differentiates between sound marriage and fornication.
"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.
Is it permissible for a woman to offer the obligatory prayer after Adhaan (call for prayer), and before Iqamah (The announcement that the prayer is about to begin) in the mosque?
Yes, it is permissible for a woman to offer the obligatory prayer immediately after the Adhan and before the Iqamah in the mosque. And Allah Knows Best.
What is the ruling on using the *subḥah* (prayer beads) to count during *tasbīḥ* and *dhikr*?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
It is recommended to use the ordinary or electronic subḥah (prayer beads or counter) to count tasbīḥ, dhikr, and istighfār, because it helps one to remember Allah. Many reports have been transmitted from the Companions and the Followers (tābiʿūn) showing that they continued to use strings of date pits as an aid to dhikr and tasbīḥ.
It is one of the lawful means of worship. Indeed, what is established from the Prophet (peace and blessings be upon him), namely that he counted tasbīḥ on his right hand, shows that it is permissible to count tasbīḥ with whatever helps a Muslim keep track of the number. There is therefore no basis for calling it an innovation (bidʿah), and we know of no jurist who said so.
Al-Ḥāfiẓ al-Suyūṭī (may Allah have mercy on him) wrote a dedicated treatise on the subject titled al-Minḥah fī al-Subḥah, printed within al-Ḥāwī li’l-Fatāwī (vol. 4–6).
Shaykh al-Islām Imam Ibn Ḥajar al-Haytamī (may Allah have mercy on him) was asked: Does the subḥah have a basis in the Sunnah? He replied: “Yes, and al-Ḥāfiẓ al-Suyūṭī wrote on this. Among the evidence is what is authentically reported from Ibn ʿUmar (may Allah be pleased with them both): ‘I saw the Prophet (peace and blessings be upon him) counting tasbīḥ on his hand.’ Ibn Abī Shaybah, Abū Dāwūd, and al-Tirmidhī also narrated: ‘Hold fast to tasbīḥ, tahlīl, and taqdīs, and do not be heedless, lest you forget the affirmation of Allah’s oneness (tawḥīd). Count them on the fingertips, for they will be questioned and made to speak.’ Counting tasbīḥ with pebbles, date pits, and a knotted string has also been reported from a group of the Companions and those after them. Some scholars held that counting on the fingertips is better than the subḥah, based on the ḥadīth of Ibn ʿUmar. Others distinguished, saying: if the one reciting is safe from miscounting, counting on the fingertips is better; otherwise the subḥah is better.” [al-Fatāwā al-Fiqhiyyah al-Kubrā, Vol.1/p.152]. And Allah Almighty knows best.