If someone delays making up fasts for a year, can they give fidyah before performing qada?
Yes, it is permissible to give fidyah before making up the missed fasts (qada), because each is an independent obligation, and there is no required order between them.
What is considered a woman`s Awrah (parts of the body that must be concealed during prayer) while offering prayer, and is her foot part of it?
A woman`s whole body is considered an Awrah while offering prayer except the hands and the face. Thus, her foot is an Awrah as well. And Allah Knows Best.
What is the ruling of Islamic Law on someone who wakes up in a state of janābah (major ritual impurity) near the very end of the prayer's designated time, during cold weather — should he heat water in order to perform ghusl, even though the prayer's time will expire as a result, or should he perform tayammum instead?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
If the water is cold, and cannot be used due to fear of harm that is not ordinarily tolerable, or that would result in illness, then in this case it becomes obligatory to heat the water — even if heating it causes the time for Fajr to expire — and no sin is incurred in doing so, due to his valid excuse. He may not resort to tayammum (dry ablution) instead, since water remains available [and merely needs to be heated].
Imam Shihāb ad-Dīn Aḥmad ar-Ramlī (raḥimahullāh) said: "He was asked regarding a person who, if he were to use cold water, would experience harm that permits tayammum, and who has access to something with which he could heat the water — but if he heats it, the prayer's time will expire. Is it obligatory for him to heat it even if the time expires, or should he pray with tayammum within the prescribed time? He answered that it is obligatory for him to heat the water, even if the time for that particular obligatory prayer expires." [Fatāwā ar-Ramlī, Vol.1/P.98]
Imam al-Bājūrī (raḥimahullāh) said in his Ḥāshiyah on Sharḥ Ibn Qāsim (Vol.1/P.173): "And if he fears [harm] from using cold water, and is currently unable to heat it, but knows of a place where firewood is available, such that if he were to go there he would not return until after the prescribed time has expired — what appears [to be the correct position] is that it is obligatory for him to go and seek the firewood, even if the time expires as a result, as this is the position ultimately settled upon by ar-Ramlī, and as related from him by Ibn Qāsim." And Allah, the Exalted, knows best.
What is the ruling on vomiting, and does it invalidate ablution?
Vomit is impure (najis). Its exit is not considered one of the nullifiers of ablution. However, the mouth must be washed and purified from it, and any that gets on clothing or the body must be washed for prayer, because prayer is not valid with impurity present on the body or clothing. And Allah the Almighty knows best.