Does a full freezer freeze items faster than an empty one? Let's say we want to freeze a banana. Would it freeze faster if we insert it into an empty freezer, or into one that already contains many frozen bananas?
 A: Heat Capacity
You are comparing two situations, and to formalize this, let's say that the freezer has a given volume, $V$ and that volume is broken down into air and other frozen materials.  Then we have the situation that $V=V_{air}+V_{stuff}$.  The heat capacity of the entire freezer between the two situations will be vastly difference because the volumetric heat capacity for air and ice (which is the most common item included in "stuff") is vastly different by a factor of about 1000.  I'll use the Wikipedia notations, where specific heat capacity is $c$ ($\frac{J}{kg K}$) and volumetric heat capacity is $c \rho$.  The volumetric heat capacity times volume is the heat capacity (denoted $C$) so for the entire freezer we have the following.
$$C = \left(V c \rho \right)_{air} + \left( V c \rho \right)_{stuff}$$
The heat capacity matters because should you insert some mount of heat into the freezer, $Q$, then after equilibrium is reached, the temperature will raise by:
$$\Delta T = \frac{Q}{C}$$
So if we don't consider the work of the freezer thermal cycle actively cooling the air, then the freezer with more stuff in it will cool the item faster because the temperature of the air+stuff changes less, so the $\Delta T$ in the Newton cooling model will be greater and it will cool faster and cool to a lower temperature.
Air Cooling
It is relevant to note that the air is actively cooled by the freezer systems and the stuff is not.  This is interesting because it will have relevance to the heat transfer mechanisms (more later).  Although this isn't entirely correct, we could assume that the cooling device takes some flow rate of air continuously, $\dot{m}$ (kg/s), and cools with some enthalpy change, $\Delta h$ (J/kg).  That enthalpy change removes heat from the system, but that doesn't change from one case to the other.
But that can't be the full picture.  Heat is only removed from the freezer if the temperature drops below a certain value.  So, another way to do this may be to assume that the cooler keeps the air at a certain temperature, but that may defeat the purpose of the problem.  It is also possible that the thing being cooled, in fact, outpaces the cooling device and heats the air faster than it can be cooled.  In that case the air temperature would rise above the programmed setpoint and stay there until the item being cooled started to approach the freezer temperature.
Heat Transfer Mechanisms
There are two mechanisms of heat transfer between the item you are freezing and other stuff in the freezer.


*

*Convection with the air

*Radiative heat transfer with other items and the walls

*Conduction with the bottom surface (same for both and I won't discuss)


Obviously, the 1st one transfers with the air and the 2nd one transfers with the stuff.  I would like to propose several simplified models such that we can talk more specifically about predictions.


*

*The freezer cools the item faster than the active cooling is relevant

*The item quickly equilibrizes with the air, then the stuff and the active cooling then slowly lowers the temperature 

*The air is kept at a constant temperature


As I've said already, number 1 clearly has the freezer with more stuff win.  Number 2 could potentially go in the direction of the freezer with less stuff because of the larger heat capacity of air (which will only work for a small item), but the radiative heat transfer still advantages the case with more stuff.  In number 3 the freezer with more stuff clearly wins unless the flow path for convection is restricted.
The exact one that will win does depend on the specific values for the system, so that's in the hands of the experimentalist.  Generally though, the higher heat capacity and the ability to transfer heat through radiative transfer is likely to favor the freezer with more stuff in it.
A: The problem can have different answers depending on what restricts the transmission of heat. I will first make the assumption that the primary restriction is in how quickly your refrigerator can move heat. That is, I will assume that heat flow rates are not a problem and that when you put a lot of hot things in your freezer, it will immediately turn on the motor and start cooling it down.
Generally speaking given two objects with the same volume, the one that is easiest to heat and cool will be the one with the fewer number of atoms. Technically, what is happening is that heat capacity is a matter of counting degrees of freedom. The larger the number of particles, the more degrees of freedom and the greater the amount of heat that corresponds to the same change in temperature.
So to see which is quicker to heat or cool down we can look at the number of atoms in the two objects. Air is a gas and the atoms are quite far apart. Therefore, air will cool down faster.

On the other hand, suppose that your question is really about adding a single small object to a freezer that is either (a) nearly full, or (b) nearly empty. In this case we might assume that opening your freezer door changes the cold air in the freezer for warm air. Then the nearly full freezer will cool your new object more quickly because there is less new air to cool down. But since air is easy to cool, this won't be much of an effect.

Finally, it's possible that different materials will have different rates at which they cool down due to differences in the rate at which heat flows. There are three ways heat moves around, radiation, conduction, and convection. In this case, gasses and liquids typically have an advantage because they convect. Since the air in your freezer is a gas,it convects and might result in cooling an item more quickly when your freezer is empty. This would be especially the case, for instance, if the rest of your freezer were filled with an insulating solid such as styrofoam insulation.
A: What is your full freezer full of? 
Situation #1: I take an empty freezer and stick a water bottle inside, resting on the freezer's floor.
Situation #2: I take the water bottle, wrap it in a high-quality down sleeping bag, and shove all that in an otherwise empty freezer.
Situation #3: I fill a freezer with ice cubes and shove the water bottle into the center of the ice cubes.
2 freezes the water more slowly than 1, because the water bottle is insulated from the floor of the freezer.
3 freezes the water more quickly than 1, because the water bottle is in thermal contact with cold stuff on all sides, not just underneath.
If you have a freezer full of food, you can make a bottle of beer freeze more quickly by inserting it somewhere such that it's in thermal contact with frozen stuff on all sides.
A: It is a fact that a full refrigerator will lower all items below the temperature seen when the refrigerator is more nearly empty.  A simple explanation:
The thermostat controlling the refrigerator is recessed and measuring the temperature of the air in the 'fridge.  "Stuff" in the fridge will by gravity be in direct contact with either the fridge or "other stuff."
"Stuff" has a high heat capacity, and as the refrigerator tries to remove heat, it gets that heat primarily from "stuff" -- both because the stuff has more heat, and because the stuff is in more intimate contact with the cooling system (conduction cooling being a primary mechanism).
The air is actually the last thing the fridge cools off, which works fine for a control system designed for a few items resting separated.  Unfortunately this control systems vastly overshoots the desired result when you've stuffed it relatively full, leaving burst soda cans as a sad testament to the oversimplified physics model used for control feedback.
A: YES, a full freezer freeze faster. 
It's simple, all the solid items stocked in the freezer at the temperature of work, let's say 0 F, loose less temperature then the surrounding air inside the freezer.
When the no freeze item goes into an empty freezer, the air changes quickly it's 0 F temperature and takes a lot to get down to 0 F because it is bad thermal transmitter.
When the freezer is full, the air changes his temperature, but the surrounding freezes items, affects promptly the air temperature, maintaining it near 0 F until the inserted item gets freeze.
