Does Laylat al-Qadr move between the nights of Ramadan?
Laylat al-Qadr is definitely in Ramadan, and it is most likely in the last ten nights, especially on the odd-numbered nights.
For this reason, the Prophet ﷺ commanded observing worship in the last ten nights, and he himself used to dedicate those nights to worship. Many people put special effort into the 27th night of Ramadan, and they are rewarded in any case, whether they actually coincide with Laylat al-Qadr or not, because worshiping on any night of Ramadan carries great rewards.
I broke my oath and am now obligated to pay the expiation for a broken oath (kaffārat al-yamīn). Would it be valid to fulfill this by inviting the poor and needy to a meal for them to eat from?
Praise be to Allah, and peace and blessings upon our master, the Messenger of Allah.
The expiation for a broken oath (kaffārat al-yamīn) is given to the poor (fuqarā') and needy (masākīn), and it must be given in the form of ownership (tamlīk) transferred to them of the required amount — namely, one mudd of the staple food of the locality, or its equivalent monetary value. It isn`t valid to invite the poor or needy to a meal prepared for them to eat from, as this does not constitute "ownership" (tamlīk) but merely "permission to partake" (ibāḥah).
Imam al-Shirbīnī (may Allah have mercy on him) said: "What is intended is that ownership be transferred to them; therefore, providing them with lunch or dinner [as a meal to eat from] isn`t valid." [Summarized from Mughnī al-Muhtāj,Vol. 5/P.50]. And Allah the Almighty knows best.
Is it obligatory to have the intention for each day of fasting, or is one intention sufficient for the whole month?
The intention is obligatory for each day of Ramadan because each day is an independent act of worship separate from the others.
The intention must be made at night before the break of dawn, as the Prophetﷺ said: "Whoever does not intend fasting at night, there is no fast for him." [An-Nasa’i]
And he also said: "Whoever does not firmly resolve to fast before dawn, there is no fast for him." [At-Tirmidhi, Abu Dawood, and An-Nasa’i]
Whoever wakes up and eats Suhoor while mindful of fasting has made the intention. Likewise, one who firmly intends at any moment during the night to fast the next day has also fulfilled the intention.
Do ear and nose drops break the fast?
Yes, ear and nose drops break the fast because both the nose and ears are open passages to the body cavity.