MBBS

Yes, you can. You can consider IGNOU. However, you need to complete 3 years degree as you cannot do it in 1 year.

You are from Karnataka, so there are a large number of private and government medical colleges you
can get if you score above 420 in NEET.

MBBS does not exist in USA.They offer MD (Doctor of Medicine) and its duration is 4 years.

Biochemical engineering is a branch of chemical engineering that mainly deals with the design and construction of unit processes that involve biological organisms or molecules, such as bioreactor.

There are a lot of good colleges abroad for MBBS. Some of them include:
Harvard University,USA
University of Cambridge, UK
University of Oxford, UK
Massachusetts Institute of Technology (MIT), USA.
You need to decide which country & college you wish to go for.
All the best.

You can work over there and then can also ask for permanent residence-ship there. In India, you can work in a hospital or practice on your own.

You need to go through all the basic concepts from NCERT books of 11th and 12th. Basic concepts of physics, chemistry and biology need to be prepared. Try to solve old papers.

To apply for MBBS, you need to complete class 12th with PCB and also you need to appear for medical entrance.