With a 7,931 rank in CUSAT CAT, will I get an admission in SOE, CUSAT or IT branch?

With a rank of 7,931 in CUSAT CAT, getting admission to the School of Engineering at CUSAT is pretty slim. Almost equal to null. Try other colleges. Although you may get admission through the NRI Spot round. 

The required cutoff for 2021 for the different branches including IT is given below. 

In certain years, the cutoff for IT reached 10000, so it depends. You may secure a seat in spot admission. Although you will need to opt for reserved seats. You can get EC, ECE on the second campus, i.e. CUCEK. But you will have to wait till the end of the entire counseling round.