What is the ruling on staring at the Holy Ka'abah?

All perfecy praise be to Allah, The Lord of The Worlds and may His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions.

Staring/looking at the Holy Ka'abah is an act of worship even if this took place while performing a prayer, since Al-Baihaqee stated that the Prophet (PBUH) said: " Staring at the Holy Ka'abah is an act of devotion." [Sho'aab Al-Imaan]. And Allah Knows Best.

What should one who made an intention, at night, to make up for a missed fasting day, but broke his fast on the next day?

One who started making up a missed fasting day, then broke his/her fast without a legitimate reason is considered sinful, and is only obliged to make up for the missed fasting days. And Allah Knows Best.

Is it permissible to say: "O Allah, by the status of the saints and the righteous, and our master Muhammad, peace be upon him, grant us the ability to reach Ramadan"?

What is mentiond by the righteous predecessors that they used to say: "O Allah bless us in Shaba'an and Rajab and give us the ability to reach Ramadan." [Shoa'ab Al-Iman/vol.3/pp.375]. And Allah Knows Best. 

What should a person who was favored from Allah with a newborn, but couldn`t afford an Aqeeqah, do?

Aqeeqah (the sheep slaughtered on the seventh day from the child`s birth) is a desirable Sunnah for the financially able since Allah, The Exalted, charges not a soul beyond its capacity. Therefore, if the father couldn`t afford the Aqeeqah before the end of his wife`s confinement, then it isn`t due on him, and if he was able to afford it later on, then it is permissible, but if he didn`t until the child reached puberty, the latter can offer the Aqeeqah himself.