Sequentially:

This depends of the game requirements. Some games has very specifics collision detection routines; some are made of OBBs, other of a sphere and a box, e etc. In this case these problems can be solved by only using a OBB-OBB test or a simply capsule-capsule for instance. Using this loss of generality you have the positive side that is fast intersection routines. Read Christer Ericsons orange book.

Recent games needs to support basically any kind of shape, and like you said, convex polyhedras are being used in recent game physics simulations.

The most commom algorithms are the SAT and the GJK.

The image you showed is a convex version of a concave mesh. It is generally transformed in convex using the QuickHull algorithm.

SAT is slightly slower but (in my experience) is numerically stable. This is because when using GJK you usually need to use EPA to get the minimum separation distance between the objects, which leads to numerical errors.

See this thread

http://www.gamedev.net/topic/667499-3d-sat-problem/

Dirk has described the SAT algorithm in details.

Thanks for clearing that up Irlan.

Do u have any suggestions on the best way to create a convex hull from an object cuz that is still something im quite confused about. I have tried to look stuff up but some of it is a bit confussing. Ill prob try and start with a simple shape like a box

You can start with a simple shape like a sphere or AABB, then switch to a OBB. Once you're familiar start looking for polyhedras, etc.

Personally, I didn't had enough time to implement efficiently the QuickHull algorithm. However you can find great slides about it in this link:

http://box2d.org/downloads/

Look for Implementing QuickHull by Dirk Gregorius. He explains also the basics of the SAT algorithm. Erin Catto explains greatly the GJK algorithm in 2D which is mathematically similar to 3D.

I have a naive implementation of the GJK-SAT algorithm (a SAT version of the GJK) here if you have trouble:

https://irlanengine.wordpress.com/downloads/physics/

In the case you have trouble with the SAT:

https://github.com/irlanrobson/Bounce/tree/master/Src/Collision

These links are highly recommended. As long you keep the authors name on the sources you can use in your projects. You should take a look on it and try to understand them.