Am I permitted to perform more than a prayer within single ablution?

All perfect 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.

You are permitted to pray as many as you could within one ablution and to recite Quran so long as your ablution isn't nullified. And Allah Knows Best.

Is a person who is in a state of Janabah (major ritual impurity due to having a marital intercourse, ejaculation, menstruation, and post-delivery impurities) sinful if he/she goes about his/her daily life activities in that state i.e. without making Ghusl (ritual bath)?

It goes without saying that a Muslim should always be in a state of ritual purity so as to be able to perform prayers and recite Quran. It is from Sunnah (Prophetic tradition) that a Muslim hastens to make Ghusl from Janabah, but he/she is not sinful in case he/she delayed that provided that he/she doesn`t miss prayers. However, it is permissible for him/her to go about their daily activities while in a state of Janabah, but had better bathe in order not to miss any prayer.

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 is the ruling on making up for missed fasting after the second half of Sha`ban (the month before Ramadhaan)?

One is obliged to make up for missed fasting before the start of next Ramadhaan, and regardless of offering it during the first, or the second half of Shab`an. This is because the prohibition mentioned in the Hadith is for offering absolute voluntary fasting in the second half of Sha`ban. And Allah Knows Best.