Does burping (belching) break the fast?
Burping does not break the fast. However, if anything comes up from the stomach, the fasting person must spit it out and rinse their mouth to cleanse it.
If they swallow it intentionally while being able to expel it, or if they swallow their saliva before rinsing their mouth, their fast is invalidated.
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.
What is the ruling on performing the Duha (forenoon) prayer before its time has begun, because I go to school early in the morning?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
The Duha (forenoon) prayer is not valid if performed before its time has begun. Its time starts when the sun has risen to a noticeable height, that is, approximately fifteen minutes after sunrise, and it ends when the sun reaches its zenith (zawāl), shortly before the Dhuhr (noon) prayer. It is stated in Mughni al-Muhtaj (Vol.1/p.455): “Its time is from the rising of the sun [to a spear’s height] until the zenith.”
Accordingly, the Duha prayer is not valid if offered before the sun has risen, and its time remains open until just before Dhuhr. It may be performed at any time at school, during the break between classes or during recess. And Allah Almighty knows best.
"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.