CCNA

Some best institutes are:
Network Bulls
Aptron
Uninets
NG Networks
Good luck.

Some of the best institutes in Delhi are-
Aptron
Network Bulls
Netexperts
UniNets etc.
Good Luck.

It all depends on your job role and career aspirations. If you are working on Microsoft technology, then obviously you must focus on Microsoft Certifications. To start with, you need to start with a single certification at a time. You cannot sail in multiple boats simultaneously. If you want to go for Cisco path then start preparing for CCNA and don't get distracted. First become expert in one domain and then when you are expert in one domain then exploring multiple domains.
All the best.

Government exams and the IT certifications you mentioned are different things. Understand the fact that government exams are aptitude based. You will have to prepare Maths, English and reasoning.
If you are preparing for government exams and in your spare time you think of doing these certifications then go for it. But, if you think that your certifications will help you in exams then it's a wrong notion.
Yes, you can go for government exams till the time you are eligible for them.

You can get a job on the basis of your skills and diploma but it will not help you in a long run. A diploma can fetch you a job for the time being but in order to grow, you need to have a degree. CCNA is a certification which will obviously increase your employability but a degree would be required to get in to good companies and grow