A shrine is a holy or sacred place, which is dedicated to a specific hero, martyr, saint, or similar figure of respect, at which they are venerated.
The direction of Karbala from India is very near to the direction of Qibla from India. You can refer to local Ulama in your area for more details.
"…Said those who prevailed in the matter, "We will surely take [for ourselves] over them a masjid." Quran 18:21
We can understand from this ayah, about the people of the cave, the permissibility of building masajid at burial sights of respected individuals.
Also, refer to Kamil al Ziyarat for many narrations about the significance of visiting Imam Hussain's (as) shrine.
No, it was not shifted. Ruqayyah Bit Al-Husain (AS) passed away in the same place where her grave is now. Yazeed put the captives from Ahlul Bayt (AS) when they reached to Shaam in a ruined ground not too far from his palace in Damascus. Ruqayyah was crying for her father Imam Husain (AS). Yazeed sent to her the head of her father covered in a tray. She got deadly shock when she saw the head of her father and passed away. She was buried in the same place.
Yes, there are many authentic Hadeeths from the Prophet (SAWA) and Ahlul Bayt (AS) encouraging building their shrines. In Kitab Al-Tahtheeb ( V.6, P. 22 ) which is one of our most authentic books of Hadeeth, narrated by 'Omara Ibn Zaid, from Abi 'Aamir the preacher of Hijaz who sai: I went to Abi Abdillah Jafar Ibn Mohammad ( Al-Sadiq) (AS) and asked him about the reward of visiting and building the shrine of Ameerul Mo'mineen (AS). Imam Jafar Al-Sadiq ( AS) narrated from his father, from his grand fathers, from Imam Ali, from the Prophet (SAWA) great reward for those believers build the shrines of Ahlul Bayt (AS) and visit them. They will get special intercetion Shafa'ah of the Prophet and Ahlul Bayt and will be with the Prophet in the Paradise. Those who help in building the shrines of Ahlul Bayt (AS) will get the reward of helping the Prophet Sulaiman in building Baitul Maqdis holy Masjid.
There is another Hadeeth in this meaning in Al-Tahtheeb, V. 6, P. 107.
I am not personally aware of narrations specifically encouraging the visiting of those sites. (Someone may correct me!) However, there are ziyarat texts narrated for them.
However, there are narrations encouraging the visiting of all of the Imams, which would include them, and it is good to go and pay respects as a gesture of loyalty and to help maintain a tie with the history and people who once walked there and were laid to rest there. Some people might also want to go as a form of resistance to the oppression that Shi'a face today and to make a stand to show that we will not be afraid to go, and that is also good.
That being said, there are a lot of narrations emphasizing visiting other places, such as Mecca, Medina, Karbala, Najaf, and Mashhad, so I would personally place the emphasis on visiting those places first.