Does using a respiratory device (oxygen) affect the validity of fasting?
Using an oxygen device does not affect the validity of fasting because oxygen is a gas with no physical substance.
However, if medicinal substances with physical particles are added to the oxygen, its use would break the fast, as it enters the body cavity through a natural passage.
What should a person, who doubts the validity of his Wudu, or over performs it, do?
One who doubts the validity of his/her Wudu while performing it should redo it until he/she is certain of having attained purity. But, if doubt was after having performed Wudu, then he/she should not pay attention to that as doubt after the end of an act of worship does not count. This is of course in case doubt was within the reasonable limits since once it goes beyond that, it becomes a whispering of the Shaytaan (devil) which he/she should ignore as the Wudu is valid.
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.
What is the ruling of Islamic Law on a latecomer who joins the imam during the standing position of the first rakʿah but was unable to complete the recitation of al-Fātiḥah?
All praise is due to Allah, and peace and blessings be upon our master the Messenger of Allah ﷺ.
If a latecomer (masbuq) joins the prayer while the imam is standing, he should begin reciting Sūrat al-Fātiḥah immediately, without pausing to recite the opening supplication (duʿāʾ al-istiftāḥ) or the seeking of refuge (taʿawwudh). If the imam bows before he completes al-Fātiḥah, he follows the imam into the bow and leaves whatever remains of al-Fātiḥah — the imam bears it on his behalf.
It is stated in ʿUmdat al-Sālik (p. 47): "If a latecomer finds the imam standing and is confident that he has enough time to recite the taʿawwudh and al-Fātiḥah in full, he may do so. If he is uncertain, he should neither recite the opening supplication nor the taʿawwudh, but rather begin directly with al-Fātiḥah. If the imam bows before he completes it, he follows him into the bow — provided he had not already begun the opening supplication or the taʿawwudh. If he had begun either of them, he continues reciting al-Fātiḥah for as long as he spent on them." And Allah the Almighty knows best.