Colleges in Jaipur

Yes, this is a registered organization.

Yes, students are eligible.

The best needs definition in terms of many criteria of colleges like infrastructure, teaching methodology, campus, facilities, placements etc. UEM Jaipur fulfills almost all criteria among private colleges in Jaipur and to know about college better with the admission process visit the link mentioned here https://www.shiksha.com/university/university-of-engineering-and-management-jaipur-47406/courses/engineering-s

A good college in Jaipur for B.Tech is UEM Jaipur, but I am not sure about the lateral entry which you can confirm from the link mentioned here:
https://www.shiksha.com/b-tech/course/b-tech-in-mechanical-engineering-university-of-engineering-and-management-jaipur-236802

Yes, as vikash has mentioned I also suggest for UEM Jaipur the one of the best you find for B.Tech in CSE https://www.shiksha.com/b-tech/course/b-tech-in-computer-science-and-engineering-university-of-engineering-and-management-jaipur-236783 and you can know more about the college and course structure through visiting the link mentioned here.

There is a minor difference in CSE and IT. In IT purely software subjects will be there but in CSE there will be software as well as hardware. So don't get confused, choose anyone. All the best.

Yes, B.Ed Degree is valid in anywhere in India for any govt exam. It doesn't depends on PTET.
All the best.