Hi

Answer to your later question is yes. You can get a seat through the JoSAA Counselling in the Government Colleges. For JoSAA Counselling you have to be very careful while filling in the choices. 

Few points I would like to mention which might prove helpful for you. Begins below - 

1. Do not fill unnecessary choices like Colleges/Branches which you might not get in any of the seven rounds. Check which you will get by tallying with the previous year closing ranks in the respective colleges.

2. Foremost thing, see which colleges/branches you can get. And make a list as per your preference. 

3. Preference can be done on the basis of the College you want to join or the Branch of study you would like to learn.

4. Preference also includes the location of the college. For example, many students avoid going to the extreme north or north east NITs and other GFTIs.

5. Fill only those choices which are favourable to you. Because if you fill in those, like a college you might not to take admission in or a branch you don't want to study. By chance if you fall into that, you have to either take admission there or opt out of the entire JoSAA Counselling Procedure.

6. Also as you must have a Home State Quota, so if you wish to study in any of the college there, you can enlist them in the top positions of your list, because chances are generally high when you utilise the Quotas. 

I hope you can fill your choices appropriately now.

Best of Luck.

Hope this helps.

Cheers!