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.

Is the one who doesn`t offer prostration of recitation during Quranic recital considered sinful?

Prostration of Quranic recital is a Sunnah due on whosoever recites a verse that contains a prostration. It is also a Sunnah for the reader and the listener. However, none of them is considered sinful for not offering it, but they deprive themselves from a great reward. It is obligatory to offer the prostration of Quranic recital in congregational prayer following the lead of the Imam.

Is a woman`s neck an Awrah (private part)?

All Perfect Praise is due to The Lord of The Worlds.

It is an Awrah, and it is forbidden for her to reveal it before non-Mahrams (Marriageable men). And Allah Knows Best.

After I had prayed, I found a drop of menstrual blood and was unsure when that happened; what is the ruling on my prayer and fast?

All praise is due to Allah, peace and blessing be upon the Messenger of Allah.

Your prayer and fast are correct ,and you should not make up for them, and we ask Allah, The Almighty, to duely reward you for these acts of worship. Your menses start once you see blood because the situation remains as such unless there is certainty or predominant conjecture, as per the Fiqh maxim: "Certainty is not removed by doubt". And Allah Knows Best.