Probably minimum required NATA scores & Jee mains paper 2 score to get Admission ?

It depends upon the JEE Main and JEE Advance cut offs, If You manage to qualify the exam by achieving the cut off, You can apply for JOSAA Counselling. It should be noted the cut off for EWS/SC/ST/FEMALES/OBC category is lower than the general category
However It is usually advised to fill the choices as per the preference of colleges and departments, for example CSE in NIT Trichy is your priority followed by CSE in NIT Warangal, you can fill CSE at NIT Trichy at serial number 1 and CSE at NIT Warangal at serial number 2 and so on and so forth. However to get any NIT, you must have qualified JEE Mains and to get any IIT, you must have qualifies JEE Advanced

But the previous cut offs of colleges should be taken in consideration, it is better to research over the cut off of colleges then make a preference list of colleges and fill accordingly

For the college preferences and the courses provided by these colleges and to know which colleges participate in the JOSAA counselling, please go to the below mentioned link: https://engineering.careers360.com/articles/josaa-participating-institutes/amp

For college prediction as per your marks and rank, please go through the below mentioned link:

College Predictor 2021 - Predict Colleges for Engineering, Medical, Law and More

College Predictors – Know your College Admission Chances Based on your Rank/Percentile, Home State, Caste, Gender, Exam Result & Get your Pe ...

Rest for information on any particular college, you can dig their official site for better information