Dual

You can check the status of any management & technology college on the website of AICTE. If the college is affiliated to AICTE, then it's a valid college.

Yes, you can get admission but for that you need to give CAT exam and score well. You can get admission in IIM – S and K. You can give other exam like CMAT, MAT, ATMA, etc. and if you get good scores in any of them, then you can get admission in any top 30 b-school.

Integrated law course is hard to find but you will find dual degrees such as LLB & Juris Doctor, LLB (Dual award).
Undergraduate law programs are different all over. In US it is called a JD program which can be done only after you have a Bachelors degree. In UK and other countries it is called Bachelor of Law or LLB. There are several dual degrees after you complete your Bachelors degree such as JD/LLM, JD/MA etc. Yale University, Harvard, London School of Economics, King's College London, Universities of Oxford and Cambridge are some of the well known universities for legal studies. Check the college websites for program options.

III Indore integrated MBA is an excellent choice for management studies after +2 and you will get similar jobs which any other PGDM graduates get from IIMI.