Question: What are the benefits of cold and hot air intakes and in specific, what does the intake manifold do?  Lastly, how should I change (if at all) my exhaust system?

Answer: The first thing that we should tackle is the Intake manifold

The intake "manifold" by definition is a pipe or chamber branching into several openings. The term is used to describe both the exhaust and the intake system of an engine. The exhaust as it exits each of the cylinders is channeled down a tube for each cylinder to a common collection point and then piped away from the engine by the tube of the exhaust system. This collection of tubes and the part where they meet is the manifold

On the intake typically the air as it enters the engine needs to be distributed to each of the engines cylinders from a single point of entry. In a car the air is usually brought in through filter system that removes any debris (like rocks or dust ) and then to the engine through some sort of metering device (throttle body ) that is hooked to the gas pedal to control how much air that the engine gets (this is usually coupled to the fuel delivery system in some fashion ) from the air metering device the air needs to be evenly distributed to each of the engines cylinders -- by "the intake manifold "

The influences of temperature
What creates the horsepower in an engine is the mixing and burning of fuel and air inside the engine -- in simple terms the more oxygen that is in the air the better burn you get.  Denser air has more oxygen -- there are more molecules of everything that makes up air in denser air (including oxygen ) Cold air is denser therefore cold air makes more power.  Therefore if you have a cooler air supply to the intake manifold of an engine then it will help the engine make more power

Hot air has less density therefore less oxygen (think of politicians giving out a lot of hot air that lacks substance ) and therefore does not make as much power. Then why would anyone have a hot or heated air supply to an engine ?

Hot air intakes are used to combat another natural occurring phenomenon:  all air has moisture in it.  At certain times of the year this moisture condenses out and forms dew on the grass or in colder times frost.  The quantity or amount of moisture in the air at anytime time affects at what temperature the moisture molecules collects together as water droplets this magic temperature of the point of condensation of the water droplets in any given air mass is called the dew point. When the dew point and the air temp (ambient temp ) get close together then you get the potential of dew or frost forming  and if you physically transform the air in anyway like jam it through a venturi of a carburetor of an engine you can induce the condensation effect and you can actually get a frost build up on surfaces inside the carburetor  -- hence the phrase carb ice.  The carburetor can actually freeze up and stop working -- if you supply even a little bit of warm air to the intake you can prevent the ice from forming and the engine will function normally.

The exhaust system: To change or not to change
Most stock exhausts are built to pass stringent noise laws and volume sound requirements by the auto manufacturers. This doesn't mean that the exhaust will produce the maximum performance from the engine -- however every car model is different and every manufacturer has different requirements and a different agenda. Most of the time a free flowing exhaust will help the engine make more power but it depends on which car and whether the engine control can adapt to the change in flow that this new exhaust could potentially create. There are a lot of good looking exhaust out there that do little or nothing for performance -- they just look nice. There are also those that change the sound and again do nothing for performance. There are true performance exhausts that make more power and are not any louder than a stock exhaust and there is everything in between these options. Whether to change or not to change I think the answer is in analyzing the purpose or goal in changing the exhaust and whether there is a product out there that suits that goal - - but it is "buyer beware" -  there is a lot of junk for sale that doesn't do what the advertising says it will.