Hello,

NEET state quota has 85% of seats reserved for the students of the state in the medical colleges whereas 15% quota are 15% seats in colleges for students outside the state (across india).{except in jammu & kashmir and telagana}
Yes you can apply for both 85% state quota of your state and 15% all india quota.
However you can book a seat by verification of documents and paying fees only through one of them. So that the students after you in the NEET rank list also get a fair chance.

You can use carrers360 college predictor to predict the colleges you can get at your rank:-

Hope this helps