B.Tech Eligibility

No you are not eligible for the B.Tech. PCM subjects are included in the syllabus of B.Tech also but biology is the main subject for B.Tech but you can get admission in engineering.

Must have mathematics, physics and chemistry in 12th class and marks should be above 60%.
(Entrance exam or direct admission)

For doing B.Tech in any stream, you should have physics, chemistry and maths in your 12th class (min. 60% in +2 ). This is the basic criteria in almost all colleges.

You may sit in the placement process with your class 10th and graduation marks as your aggregate score would be in the 60s. However, there is a zero-day in the placement season when 13-14 companies come to campus which does not ask for any eligibility criteria. You can seize that day my friend.

Yes, outside students can apply. But you have to give either JEE or GUJCET exam to take admission in here because admission to BE is done on the basis of rank secured in Gujarat Common Entrance Test GUJCET conducted by Gujarat Secondary and Higher Secondary Education Board (GSHSEB) or Joint Entrance Examination (JEE Mains) conducted by Central Board of Secondary Education (CBSE). 60% of JEE marks and 40% of boards marks are taken into consideration for the ranks.

No, it is highly impossible for you to get a free seat at this Rank. You may get into your desired college through management, but with rank>100000 free seat is impossible.

You are eligible for B.Tech courses. Your rank is very good and I think there is some chances of getting good college. Checkout the link below to find the college you are eligible for :
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor?utm_source=connect&utm_medium=CR_JEECP&utm_campaign=6540165.