(Hazrat)= As respect like Sir or Dr. or Mr.
(Khwaja)= Old or respectful.
(Moinuddin)= Guide of Islam.
(Hassan)= Real Name is Hassan only.
(Chishty)= A culture where people like music and do no limit charity.
(Sanjari)= A village of Iran birth place of Huzoor Gareeb Nawaz (R.A.).
(Summa)= here also from.
(Ajmeri)= A city where Khwaja Garib Nawaz (R.A.) Dargah is located now.
(R. A.)= R. mean Rahmatulla Allah's mercy on him and A. Aleh mean him. 

Yes according to this Maternal Shijra (A family tree):
Bibi Ummul Wara (R.A.) also known as (Bibi Mahenoor (R.A.))
D/O Hazrat Syed Daud (R.A)
S/O Hazrat Abdullah Hambli (R.A)
S/O Hazrat Zahid (R.A)
S/O Hazrat Mohammed Mauris (R.A)
S/O Hazrat Dawood (R.A)
S/O Hazrat Musa (R.A)
S/O Hazrat Syed Abdullah Makhfi (R.A)
S/O Hazrat Hasan Masna (R.A)
S/O Hazrat Syedna Imam Hassan (RaziAllahoTala Anho)
S/O Hazrat Amerul Momeineen Mola Ali Mushkilkush Sher -e- Khuda (KaramAllahuTala Wajihul Kareem)  

Yes according to this Paternal Shijra (A family tree):
S/O Hazrat Gyasudin (R.A)
S/O Hazrat Najmuddin Tahir (R.A)
S/O Hazrat Syed Ibraheem (R.A)
S/O Hazrat Syed Idrees (R.A)
S/O Hazrat Imam Musa Kazim (R.A)
S/O Hazrat Jafar Sadiq (R.A)
S/O Hazrat Mohammed Baqar (R.A)
S/O Hazrat Zainul Aabedeen (R.A)
S/O Hazrat Syedna Imam Husain (R.A)
S/O Hazrat Amerul Momeineen Mola Ali Mushkilkush Shere Khuda (KaramAllahuTala Wajihul Kareem)  

His mentor was Khwaja Molana Hisamuddin Bukhari (R.A.) who gave him his own Jhubba and Dastar as an education certificate. Yes he was educated and he completed his education in all available sources of knowledge. He had all the titles of that time which use for an educated person some of the titles are below.
Kari,
Muhaddis,
Shaikh,
Mufti,
Allahma,
Molan,
Hafiz and all other titles which use for Muslim scholar and he was a poet also.

Yeah that is true some arabic words which wrote by Jibraeel (A.S.) as a sign that this is Allah's man Allah took him back. “Haaza Habib-Allah Mata Fee Hoob-Allah”  As a signature of Allah (Rabbul Izzat) these words was wrote on fore head of Hazrat Khwaja Moinuddin Hassan Chishty Sanjari Summa Ajmeri (R. A.).

Chhati is just a date which is 6th of every month according to Muslim calendar. But that date is very important for Chishty culture because they celebrate this date as wedding of Khwaja Garib Nawaz (R.A.). Now the point is how they celebrate wedding on every month. So the answer is (Urs) which is an Arabic word which come from (Urusi) which use for human marriage where two people bound in a relationship. But the (Urs) is slight different from this word. (Urs) mean completed the connection with Allah called (Urs) where the person who's (Urs) is that can direct interact with Allah Rabbul Izzat. So people celebrate that date when Khwaja Garib Nawaz (R.A.) die and made a direct contact with Allah (The creator). For death history click here.