JNTUH - Jawaharlal Nehru Technological University, Hyderabad

Yes, you still have a chance in getting a seat in IDP (B.Tech +M.Tech ) in Mechanical Engineering or Civil Engineering .Also, don't forget to check with EEE (IDP) also. This year, there may be a chance.

No other examination scores are needed, you only need TS EAMCET score.

For this rank, around 5000-6000, yes you have 80% chances of getting a seat in JNTU-Hyderabad.Besides, having a rank around 5000-6000, if you are an SC/ST candidate, you may also get a seat for CSE (Reg), this is a four-year course of graduation and CSE-IDP is a five-year integrated course.
You may choose this based on your interest.

Are you sure that your rank lies around 15,000 (TS EAMCET)?
If so, there may be a chance for you to get a seat in mechanical (reg/idp).
But the allotment of seats is also based on your caste.
If you come under BC/SC/ST, there may be a chance for you.
But if you are an OC (general candidate), then it is not possible, your rank should be under 5000.

Yes, JNTU offers admission to Diploma students also, the one who is allotted seat based on his/her ECET rank can directly join in the second year with regular (4-year course) students.

Till now aeronautical is not offered in JNTU. Try for another colleges.