Engineer eligibility

First, if you have completed your 3 years diploma in mechanical engineering, then you are eligible for various state and central government jobs. Keep a close watch on leading job portals and employment news.
Second, for boiler operation engineer, you must meet the following criteria-
i) 3 years diploma or BE/Btech in Mechanical, Electrical, Chemical, Instrumentation, Power Plant and Production Engineering.
ii) Candidates should have experience in Operation and/or Maintenance of Boiler for
a) Not less than two years in case of Degree holders
b) Not less than Five years in case of Diploma holders

No, there is no examination available for engineering now. You can give state level exam or you can switch to private university.

BE/B.Tech or equivalent qualification with an aggregate minimum of 65% marks or CGPA 6.84/10 (average of all semesters for which results are available). Minimum of 65% marks or CGPA 6.84 in section B alone for candidates with AMIE/graduation IETE qualification. Candidates who are going to complete the above course in the academic year 2016-17 are also eligible to apply provided final degree is available by 31/8/2017. The qualification prescribed and the benchmark are only the minimum requirement and fulfilling the same does not automatically make candidates eligible for written test.

Is Biology not a reasonable prerequisite to expect from future biomedical engineers? Of course it is an appropriate expectation from the engineers, since they are designing for the human body.

Yes, you are eligible for engineering.

To get admission in any IITs, you first need to crack JEE Mains paper-1 and then JEE Advanced and you need atleast 75% in boards to be eligible to get admission in any IITs.