Hi thani, 

The mop up rounds of karnataka and tamil nadu will take place in the mid of August.  You will  be notify on the official site of their counselling organization. 

Just keep on checking their sites and stay aware and updated. 

Yes karnataka is only state that allows non domicile students to participate in their state counselling.  So you can participate in that but you have to register for its document verification many days before.  If you have not registered then you can't participate now. 

For any other information you can ask us. 

Hello dear student, 

The rank you have got is really far and trust me there are almost no chances that you get admission in any government medical college in mop up round .  Cut offs are really very high and students have gained such SKy rocketing marks that competition have been increased so much.  I hope you understand it's really difficult .but I am sharing a link with you kindly go through it once 

https://www.google.com/amp/s/medicine.careers360.com/articles/neet-cut-off-for-government-colleges/amp