What would happen if the second law of thermodynamics was not there for governing in every day processes. Can a universe exist and evolve if the second law of thermodynamics does not exist in first place?

There is a universe where the second law doesn't hold. In fact, the opposite holds: entropy always decreases.
Things are a bit strange there, though. The people there can easily predict the future many years in advance (although they don't call it "predicting" but rather "gnirebmemer"), but even their best scientists can't remember what the weather was like more than a few days ago. (They call it "gnitciderp"). Instead of being born, people are dug up out of the ground, and the only way anyone can ever die is to be put inside their mother's body, where they gradually shrink down to a single cell and then, eventually, nothing.
This world, of course, is our world, only with the direction of time reversed. On the level of atoms and photons, the laws of physics can be reversed in time, so in a sense this is a universe that really could exist. In a deeper sense it actually does exist, and we live in it. So it seems the second law is not necessary in order for the universe to exist, but rather, it's an empirical fact that we have to explain. The usual explanation is that the entropy is low at one end of time (the big bang) but not at the other, so it has to form a gradient between the two. (Though admittedly this does sound a bit circular when put that way.)
For further reading you can look into "the arrow of time" and "Loschmidt's paradox". Unfortunately the Wikipedia articles are not particularly well written, but with a bit of searching you should be able to find some good resources. If you want a book, I'd recommend "Time's Arrow and Archimedes' point" by Huw Price, although some of it's very technical and the cosmology parts are out of date now.