BCA

BCA degree from any recognized university or institute is a valid graduation degree for higher studies and job in both government and private sector.

As such, there is no entrance exam for integrated BCA +MCA but there are various exams of different colleges and universities for this course.

Yes, it is true because a lot of students prefer pursuing MCA and engineering. Do you still think that companies will still prefer BCA graduates over MCA and engineering students? You should better go for a Master's degree course.

Yes, any recognized university's degree is valid to appear for UPSC exam.

Some universities syllabus may have maths or statistics as a subject but something difference also will be available in the extra subject for BCA and B.Com. Mostly (except for B.Com) BCA and B.Sc in maths will be totally same in syllabus and subject pattern because both the degrees belong to science stream and for B.Com, the maths will be different due to the commerce. For example, accounting and taxation, but the syllabus of the statistics in B.Com will remain the same as in B.Sc and BBA.

Dear,
Different Indian universities follow almost same syllabus in BCA. Here I am giving Ist sem subjects followed in MAKAUT(WBUT). There are five theory papers & two lab papers
Theory:
1.Digital electronics
2.Business systems &Application
3. Introduction to Programming
4. Mathematics
5. PC software
Two Labs:
1. pc software lab
2. programming lab

Follow the official website for all the details:
http://www.stxaviersjaipur.org/bca.html

As a diploma holder, you are eligible for direct second year entry in BCA and then, your BCA duration will be of 2 years.

There are very few study centers under IGNOU offers lateral entry for BCA. For detailed information, you can check out the official link below:
http://www.ignou.ac.in/ignou/aboutignou/school/socis/programmes/detail/190/2