Is it a condition for the mosque where I‘tikaf is performed to hold Jumu‘ah prayers?
No, it is not a condition for the mosque where I‘tikaf is performed to have Jumu‘ah prayers. However, if a person vows to perform continuous I‘tikaf, and Jumu‘ah occurs within that period, then they must observe I‘tikaf in a mosque where Jumu‘ah is held so that they do not break their continuity by leaving for Friday prayer.
Is it permissible for an elderly person, a pregnant woman, or someone with a chronic illness to pay fidyah before Ramadan begins?
It is not permissible for an elderly person, a pregnant woman, or someone with a chronic illness to pay fidyah before Ramadan begins.
Additionally, it is not allowed to pay fidyah for more than one day in advance, because fidyah is a substitute for fasting, and fasting is not yet obligatory at that time.
However, it is permissible to pay fidyah for a single day in advance, by analogy with paying zakat up to one year in advance.
What is the ruling on someone who prays but does not fast, without a valid excuse?
Fasting is one of the pillars of Islam, and a Muslim's faith is incomplete if they neglect the obligation of fasting while being physically capable of observing it.
Someone wiped over his shoes after wearing them in a state of purity, then took them off and prayed without them?
Ablution is not invalidated by taking off leather socks or shoes after wiping over them. However, whoever takes them off after wiping must wash his feet only. If he prayed without washing his feet, he must wash his feet and repeat the prayer. And Allah the Almighty knows best.