Just check the reference like so:

Core.objectCollision = function() {
    if (Core.Mouse.isHolding != null) {
        for (i in Core.Objects) {

            // Skip the object that was grabbed.
            if (i == Core.Mouse.isHolding)
                continue;

            if (isColliding(Core.Mouse.isHolding, Core.Objects[i])) {
                console.log("Colliding with world object!");
            }
        }
    }
}

just skip it inside your for loop.

To prevent the check of the object colliding against itself, just check the reference like so:

Core.objectCollision = function() {
    if (Core.Mouse.isHolding != null) {
        for (i in Core.Objects) {

            // Skip the object that was grabbed.
            if (i == Core.Mouse.isHolding)
                continue;

            if (isColliding(Core.Mouse.isHolding, Core.Objects[i])) {
                console.log("Colliding with world object!");
            }
        }
    }
}

Now, if you want to implement a physics engine, I really recommend you see this series https://www.youtube.com/watch?v=3Oay1YxkP5c even thou he implements a 3D engine, the core logic is the same.

Then, to get a better performance, you might want to look at quad trees so that objects that are far from each other are not checked for collision, this: http://buildnewgames.com/gamephysics/ should give you some directions.