Does passing wind from the anus affect ritual purity?
Ablution is nullified by the passing of wind from the anus, but washing the anus is not required as wind leaves no impurity.
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.
Which is better for a traveler: fasting or breaking the fast?
Fasting is better, unless it causes hardship—in that case, breaking the fast is better.
What is the ruling on the ablution of one who cuts his nails, and is it permissible to cut them before the ritual bath for major impurity (janabah)?
Cutting nails does not invalidate ablution, and it is permissible to cut them before the ritual bath for major impurity. And Allah the Almighty knows best.