1)What do we mean by negative and positive energies in physics?
Negative energies are lower than positive energies. Most areas of physics only care about whether one energy is lower than another and by how much.
2)I have seen in a minute physics video that we need negative energy to prevent a wormhole from collapsing. Why?
General Relativity actually cares whether energy is positive or negative, not just whether one is lower than another and by how much. Roughly positive energy makes things come together, like if you dropped balls near the earth, they all head towards the center of the earth so they get closer to each other. If you want defocusing you need negative energy. And you want defocusing if you want some small region to take curves going in and have them defocus to spread out to actually be going away from each other when coming out the other side of the wormhole. So you need it if you want curves to actually make it out through the other side.
3)Why do we need negative energy in warp drives for faster than light travel?
You need them for warp drives even if the warp drive goes slower than light. That's part of what it means to be a warp drive. It is basically surrounding something with so much negative energy that it can move in a novel fashion.
4)Overall question:- What is the role of negative energy over positive energy?
The overall effect is stability. Since it is easy to send out positive energy it is easy for things to get stuck in lower energy states by sending out some positive energy outwards. This is how planets and stars form how atoms and molecules form and how most things happen if they are stable for some time.
Having a lower bound on energy can give stability which allows structuresto persist and develop. Or course energy can flow through a system too, so its not the only thing going on. Energy is associated with dynamics and time so its pretty much what physics in general is about.