JEE

With this rank it is very difficult to get good branch in these engineering colleges although you have state quota.

Since your rank is very low, there are very low chances of getting civil branch in NIT or IIITs you can try out GFTIs.

Yes, you can get into good colleges if you didn't score well in JEE Mains. BITS Pilani, CEAU Guindy, MIT Karnataka, NSIT Delhi, MIT Pune, SRM University, MSRIT Bangalore, R.V. College of Engineering, and more colleges provide admission to engineering colleges without JEE Main.

The admission is based upon a cut off criteria thus cross check the cut off for your preferred course. While it is difficult to get it through score of 16K you may apply to multiple instituitons at the same time. 

Yes, there are several colleges and universities that offer B.Tech in Computer Science and Engineering (CSE) with comparatively lower fees. These include government or public-funded institutions, state universities, and some private colleges.

With this percentage it is very difficult to get admission in NITs or good government colleges but you can apply in good private universities or institutes.  You can also try for state level counseling for engineering admission.

No, you will not get any NIT for BTech. However I'll suggest you to opt for some private college entrance exams so that you get a good college with a good branch. 

Also, you haven't mentioned your category but if your'e from Pwd then there is some chances that you'll be getting. 

So, try for some other colleges for BTech.

All the best!

Don't lose hope, try some low level colleges like Siddhant (dehu),etc.

Yes, yyou have chances to get admission in RAIT. 

If you are asking for CSE in NIT then let me tell you, you got very good rank. You just need to fill your choices in JoSAA. I think you can get NIT.