Hello Sir. Yes, definitely you can opt for both the options. The reason is simple the dates wont clash, and you will have wider range of options.
Domicile for state level quota is indeed. For all India quota, you dont need that.
For more information, please refer careers360 article.
https://www.google.com/amp/s/medicine.careers360.com/articles/neet-counselling-faqs/amp
Hope that helps!
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