As you have mentioned, you have Delhi domicile but are willing to take admission in a private medical college in Bihar- for that you must register for the state counselling of Bihar. Only for admission in deemed medical colleges or government ones, you must register for All India Counselling.

Yes as Bihar is an open state you can apply for private colleges of Bihar being a non domicile candidate.

Here is a list of colleges you can apply in:

• Katihar Medical College & Hospital (KMCH), Katihar
• Lord Buddha Koshi Medical College and Hospital, Saharsa
• Mata Gujri Memorial Medical College and Lions Seva Kendra Hospital, Kishanganj
• Narayan Medical College and Hospital (NMCH), Sasaram