Hello,

Yes you can give JEE exam in and get admission in IIT's and NIT's. You can appear for both jee mains and advance if you appeared for 12th class in 2019. If you appeared for 12th class in 2018 you can only give mains. So you can appear according to eligiblity. But if you take admission you have to start engineering again from 1st year again.

Of course you can prepare for JEE to get IIT and NIT in 2020 along with your 1st year of engineering . 

There are no special branch subjects during 1st year except engineering graphics , C-language , workshop and labs etc but they are not much painful you can study other stuff regarding JEE also .

You can start your preparation from any renowned institute's study material or go with standard books . Start from the very beginning if you did not tried it in 1st attempt .

Focus on scoring areas and try a lot of problems 

for maths - Go for calculus, vector and 3-D , conic section , straight line, circle , quadratic equation these are among scoring areas and if time permits brush up logs , sequence and series and probability .

for Physics - soft and scoring areas are magnetism and electricity portion , modern physics , waves , thermo-dynamics , SHM etc . 
Practice H.C Verma problems and cement your concepts 

Chemistry : for Inorganic chemistry read NCERT  2-3 times and learn it by heart till the day of your exam , imp chapters are Bonding , Co-ordination compounds , surface chemistry p-block , stoichiometry , redox reactions etc. 
Practicing question bank will be a revision kind of thing in Inorganic chemistry .

Organic chem: Understand the concepts of GOC thoroughly because they cover a huge portion , questions like acidic/basic strength , stability , order of bond length , order of ionisation energy can be answered directly from here 
than study carbo-cation , carbo-anion , aromatic reactions separately
Focus the name reactions like aldol-condensation , cannizaro , holfman bromamide reaction , etc mechanism must be in mind while solving the problem 
than comes the methods of oxidation , reduction , various reaction intermediates,  chemical conversions etc . Practice any standard question bank 

Physical chemistry : do a thorough study of all the chapters including derivations , diagrams , graphs etc and try problems 1st from NCERT than go for any study material . 

Note : 2 subjects must be strong enough and chemistry is very scoring so make it your interest