Who is the one required to slaughter the 'aqīqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is Sunnah for the 'aqīqah to be slaughtered by the guardian of the newborn — the one upon whom the child's financial maintenance is obligatory — provided he is financially capable of doing so. 'Alī ibn Abī Ṭālib, may Allah be pleased with him, reported: "The Messenger of Allah ﷺ performed the 'aqīqah for al-Ḥasan with one sheep." — Narrated by al-Tirmidhī. And Allah Almighty knows best.
Is it permissible to combine and shorten prayers at home before traveling?
It is impermissible to combine and shorten prayers at home before traveling because a person is called a traveler once he/she departs the borders of his/her country, and offering prayer that way was originally intended to exempt a traveler. And Allah Knows Best.
What is recommended for a Muslim to say on Laylat al-Qadr?
On Laylat al-Qadr, it is recommended for a Muslim to recite the supplication (du‘a) that the Prophet ﷺ taught Aisha (may Allah be pleased with her):
"O Allah, You are Most Forgiving, and You love forgiveness, so forgive me." [Narrated by Al-Tirmidhi]
How do I deal with whispers (waswasa) in ablution and purification?
Whispers in ablution are from Satan, and the Muslim should not pay attention to them. He is not required to re-perform ablution or repeat washing a limb because of waswasa. He should always base his purification on its being valid. It is recommended for him to frequently say "La ilaha illa Allah" (There is no god but Allah), because Satan recoils when Allah is mentioned. And Allah the Almighty knows best.