Which lens should be selected between Nikon 55-300 DX format and Nikon 70-300 FX format lens for my D5200?Will there be any advantages in choosing FX format lens for my DX format camera?I heard that Nikon 70-300 FX lens is faster in focus than 55-300 and uses IF.Can you please help me in deciding the lens?

It depends on your shooting style.  If you like to shoot action and birds, get a 70-300.  That type of lens is faster focusing.  If you like to walkaround with a lens for general street/travel type photography, the 55-300 might be fine.  It is lighter and wider so you might find it great for everyday use.  Since the 55-300 rotates while focusing, filters are difficult to use.  So, if you are going to use polarizers and such, get the 70-300.  I don't often use filters on my telephoto so that might not be an issue for you.   I chose the Tamron 70-300 vc and think it's the bee's knees.  Your mileage may vary (YMMV).

Thanks for the info.. Will there be any diffrences seen with FX lens on DX body?

The 70-300 is a better lens full stop. Can be used fx or dx. Good fast Af.The 55-330 is lighter and has a better placed focus ring but it has noticeable aberration and handles flare badly.I had the 55-300 largely because of Mansurov's favourable review and price but I  traded it in for the 70-300 when using a V1. Much better IQ.