Civil and Structural Engineering

USA and UK are the best for structural engineering.

It depends on the individual's interest. If you are interested in the technical field, then go for ME, but if you want to work in the management field, then you can opt for MBA. In current scenario, MBA has more scope than ME.

You did not provided sufficient information here. Civil is the only branch for B.Tech and after b.tech you can go other sub branches of it in m.tech like structural engineering, environmental engineering, Geotechnical engineering etc
You just visit the website of the mentioned institute and contact them either by mail or phone for more queries.

Still structural engineering offers a better job opportunity in India, either in academia or in industry however if you have a plan to go for PhD than environmental engineering has a very bright future. This branch gets easy sponsored project as compared to structural engineering.

why you are looking for private universities?
Try some government institutions. You can go for college affiliated to UPTU or Amity, Galgotias, Shardha etc

In Abroad the universities are offering as MSc. Structural Engineering. UK it is one year masters degree.

can we become stuctural engineer after doing diploma in civil engineering