1. Who established the Islamic Year and when was it established?
Saiyidina Umar Al-Khattab established it, marking the year of the hijrah as the start of the Islamic calendar, because it was an important turning point for the Muslims. It was established about 17 years after the actual hijrah, with the consensus of his scholars and advisors.
2. What should we do at the start of the new Hijrah year?
We COULD reflect on the event of the hijrah and together try to understand the REAL meaning of Hijrah.
3. What about that do'a awal tahun akhir tahun.
As far as I know, and can (can't) see, there isn't ANY hadith mentioning this. Especially on the 'fadhilat' of doing the supplication 3 times and so on. So, doing it with the i'tikad that you'll get the fadhilat is
I imagined how it may have started. Maybe some fella asked his ustaz.. what shud we do on the start of the new islamic year. Like the Westerners, they celebrate the New Year.. can we?. So maybe the ustaz said.. we shouldn't celebrate like the westerners especially with booze and immoral behaviours. If you want to do something.. why don't you just supplicate, doa - that the past year and the coming year brings good to you.. etc.. PERHAPS when you read it, the syaitan will 'blah blah blah' (see the 'fadhilat' baca doa). And so the man did.. and passed on this to his family and children.. and VERY soon, it became a die-die-must-do tradition. MAYBE. Yes, I have wild and great imaginations.
Anyhow, if you want to continue reading the doa, just because you want to doa - not for any specific reasons, I guess it's ok to do so, but NOT because you THINK it is something expected of a Muslim on the Muslim New Year.
4. Anything special about the month of Muharram?
It is mentioned by the Prophet, to be one of the 4 sacred months, mentioned in the Quran. On the 10th of Muharram, day of Asyura, we are encouraged to fast for 2 days, 9th and 10th. This is to differentiate our practise, from the practise of Jews and Christians.
The number of months in the sight of Allah is twelve (in a year)- so ordained by Him the day He created the heavens and the earth; of them four are sacred: that is the straight usage. So wrong not yourselves therein, and fight the Pagans all together as they fight you all together. But know that Allah is with those who restrain themselves.
Al Quran, At-Tauba 9:36
Narrated Abu Bakra:
The Prophet said. "(The division of time has turned to its original form which was current when Allah created the Heavens and the Earths. The year is of twelve months, out of which four months are sacred: Three are in succession Dhul-Qa' da, Dhul-Hijja and Muharram, and (the fourth is) Rajab of (the tribe of) Mudar which comes between Jumadi-ath-Thaniyah and Sha ban."
(Sohih Bukhari Volume 4, Book 54, Number 419)
Narrated Ibn 'Abbas:
When the Prophet came to Medina, he found (the Jews) fasting on the day of 'Ashura' (i.e. 10th of Muharram). They used to say: "This is a great day on which Allah saved Moses and drowned the folk of Pharaoh. Moses observed the fast on this day, as a sign of gratitude to Allah." The Prophet said, "I am closer to Moses than they." So, he observed the fast (on that day) and ordered the Muslims to fast on it.
(Sahih Bukhari Volume 4, Book 55, Number 609)
The people used to fast on 'Ashura (the tenth day of the month of Muharram) before the fasting of Ramadan was made obligatory. And on that day the Ka'ba used to be covered with a cover. When Allah made the fasting of the month of Ramadan compulsory, Allah's Apostle said, "Whoever wishes to fast (on the day of 'Ashura') may do so; and whoever wishes to leave it can do so."
(Sahih Bukhari Volume 2, Book 26, Number 662)
IslamOnline : Umar Al-Khattab
About.com : Islam : What is the Hijrah Calendar?
IslamOnline : Can we celebrate Hijrah
IslamOnline : Fasting the 9th and 10th of Muharram
ps: This is as far as I know. Correct me if I'm wrong, advise me if I miss anything. But please provide authentic dalil and/or sources with it.
Have a great day!