College of Engineering, Pune

Yes, definitely you are eligible. You just need to find out the cut-offs for different streams.

Yes, you can apply for B.Tech in civil engineering.
I am also pursuing B.Tech in India after completing diploma in civil.

JEE Mains Cutoff of COEP varies every year, additionally from 2016 MHTCET examination will be also conducted for engineering admissions. Below are the JEE Mains branchwise (CAP I) for COEP as follows:
Civil engineering - 170
Computer engineering - 189
IT - 181
Electrical engineering - 178
E&TC engineering - 180
Instrumentation engineering - 172
Mechanical engineering - 182
Production engineering (Sandwich) - 163
Note: Above JEE Mains Merit Scores are for GOPENS category.

VIT B School is better in terms of placement, ROI, Faculty, Infra and others factor too.

You have to appear for LEET exam. The admission process for diploma holders is ranking wise. But, still, you have to score 90 to 95 percentage for appearing in COEP in final year.

COEP being a government autonomous college has a slight edge over PICT. However, there's no doubt PICT is amongst the best in Pune, if not Maharashtra.

MHCET is the qualifying exam. So, clearing minimum cutoff for boards and your main focus must be scoring MHCET. There is no link between boards and any competitive exam. So, focus on it.

You need to have minimum 60% in Diploma to be eligible for admission in engineering in COEP through lateral entry. Diploma Percentage to get into COEP is subject to number of students appearing for counselling and the cutoff set by the college
https://www.shiksha.com/b-tech/course/b-tech-lateral-entry-college-of-engineering-Pune-shivaji-Nagar-204728.