IIT Bhubaneswar - Indian Institute of Technology

Old 5 IITs > NIT TRICHY > New IITs.
If you get at any of these 5 IITs then that should be your 1st preference:
BOMBAY
DELHI
MADRAS
KANPUR
KHARAGPUR
If you dont get in the above then join NIT TRICHY.
Let kme tell you that NIT TRICHY is better than all the new IITs.
So, I would recommend you to think on similar lines and decide as to what combination might work out the best for you. One thing I would strongly advise is not to make decisions based on emotions or attachments related to an institute, parents (with reference to staying near them) or friends. I know a few people who chose IIT just for the sake of being in IIT with no regards to the branch, when they had the option of getting a much better one at a lesser institute and they definitely regretted making that choice.
I would be glad to offer any insights about NITT in general and EEE, in particular if you ever feel the need.

Civil is more of a core branch while design is not. It completely depends on your interests as to decide where you should go. Institutes do matter but branch matters a lot more.
Quality of teaching: I can tell you that civil department professors at IIT-Bhubaneswar are quite good. Although I don't know much about the professors at IIT-Guwahati but still, IIT-Guwahati being older will be equally good if not better.
Campus life: IIT-Guwahati has a much better campus life than IIT-Bhubaneswar, so there will be much better exposure. But newer IITs are improving at a fast rate. There will be fewer students in IIT-Bhubaneswar compared to IIT-Guwahati so there are better opportunities for an individual to actively participate in extra academics activities like societies and fests. Although sports facilities at IIT Guwahati are much better than IIT-Bhubaneswar.
Placements: IIT-Guwahati design will be mostly offering non-core placements while at IIT-Bhubaneswar you have a much better chance to get core placements (this year more than 70 percent students got core placements). So, if you have more interest in the technical stuff then I would suggest IIT-Bhubaneswar civil as a better choice over IIT-Guwahati design. The average package for IIT-Guwahati design will be greater than IIT-Bhubaneswar's civil branch.