Yes .

Most of the pre engineering aspirants choose kota for the preparation.  And the home state quota apply on the basis of 12th mark sheet.

Students who are preparing, apply for the 12th board in the schools of kota . That's why the students from rajasthan not get the benefit of home state https://youtube.com/channel/UCIcl-k8hbdavWKwCNeuD2gA

Yes .

The home state quota for mnit Jaipur in the opposite way.  And the major reason for that educational hub.

home state quota apply on the based upon the 12th marksheet.

And most of the  students who is preparing in kota , apply in the schools of kota for the +2 board. That's why students from rajasthan not get the benefit of home state.

Yes, This is possible.

Generally, the NITs system provides some privileges to home state students. Like NIT Patna is located in state Bihar, so according to the home state reservation, 50% of the total seats of NIT Patna is reserved for students who have a domicile certificate of Bihar, that is the resident of Bihar.

But in some cases, this rule goes against its home state. Examples of such states are Rajasthan( MNIT Jaipur ) and Uttar Pradesh( MNIT Allahabad ).

Here I will give you an idea how it looks like. Last year such thing again happened in MNIT Jaipur, not in all branches but it happened in most of the branches. Below is the 2020 cutoff(male) rank of the Chemical engineering branch in MNIT Jaipur.

Category    Home State   Other State

OPEN             22984         24475

EWS                4468           4568

OBC NCL        7298           8563

SC                   4879           5836

ST                   2149           2580

(Source: Careers360)

As you can see cutoffs close at a higher rank for other state students and it is surprisingly benefitting them in comparison to home state students.

This happens because of the following reasons:-

1. In some states like Bihar, Uttar Pradesh, Rajasthan, etc.,  there so many students look for IITs and NIITs for their brand values and people are very aware of this exam  JEE main as compared to other states. Now most of the students want to study in their home state NIT which is located for their ease of traveling, no environmental change, staying near to their family, so many choose the home state NIT as their first preference and it ultimately rises competition among the home state students.
2. In some states there are lots of good coaching centres are available then students from all over India come for JEE preparation in that state, eg.- Rajasthan(KOTA). Now as many students do their 11th and12th schooling from that same state so according to JOOSA which performs the selection process, such students are considered to be from the state from where they have completed their 11th and 12th schooling.