Hey Ishant,
With regards to your question, I would suggest that the right time to go for polytechnic is after 10th.
If you want to pursue a higher degree in engineering after 12th std you should go for Bachelors in Engineering/Technology.
While pursuing poly after 12th will be a waste of 3 years for you(or 2 years if you go for engineering after polytechnic).
And speaking of the admission procedure, it is based on your 10th boards certificate. And you will have to submit your documents for 10th and 12th boards as well as school leaving certificate for 12th std.
But I would suggest you to go for B. E. Or B. Tech.
I hope this answers your concerns.