GST Registration

Yes. You can take take registration.

You may rented building for your office address and then proceed

Addhaar & bank account is not an address proof , required to provide in GST Registration.

The new clause (4A ) has been inserted under Section Rule 8 of CGST that Applicant shall undergo adhaar authentication of Adhaar number for grant of Registration & In term of Rule 10A of CGST Rules , bank details can be submitted with in 45 days from the date of grant Registration.
Hence if you need the new bank account can be open in the state can provide details therein.

Yes, you can take registration inTelangana despite your Aadhar Address belongs to Bihar for registration you need only rent agreement as address proof of your business premises followed by Electricity Bill 

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