How does ice make room temperature water so cold so fast? Does the water from the ice melting off when the room temp water hits the ice cube make it cold?

ice absorbs heat from water.

Explanation:

Heat is a measure of molecular activity: the faster a thing's molecules move, the more heat that substance contains. And because everything is made of molecules, and molecules are in constant motion, everything contains at least some heat.

Because ice molecules move slowly and cluster tightly together, they produce a relatively low amount of heat. When ice is placed in a glass of water whose molecules are moving at a greater rate, the ice begins to absorb energy, because heat always travels from regions of relative warmth to colder areas in order to equalize temperatures. In other words, ice absorbs heat from the water.

As the water molecules lose energy, they begin to slow down, and consequently cool. So, it's kind of the opposite of what you might think: when we put ice in water, the ice doesn't give its cold to the water, it takes heat from the water.