1. What do I put the Arduino actually in? Is there some common Arduino housings that people use?
Whatever you need to protect it from whatever it needs protecting from. No, there isn't.
2. What Power Source will I need to use? Can I use 1 power source for the Arduino and the Fan?
The fan is AC. The Arduino is DC. What do you think?
3. How do I switch on 120v devices? Is there a simple cheap relay that would work for this?
There are relay kits, that include the relay and the transistor needed to control the relay. There are SSR relays that can can be directly used with an Arduino. Simple, yes. Cheap, not necessarily, although I don't think cheap should be a criteria. Inexpensive, perhaps. Cheap, no.
4. How will I adjust when I want the fan to come on (like at what temp) and would I see what temperature it is.
Depends on what hardware you want to use. Switches allow incrementing and decrementing in discrete steps. A potentiometer would allow a more uniform control. However, measuring the actual temperature is more of a concern than being able to define/adjust a set point.
Would I need and LCD screen?
If you want to see some data, then, yes.
Buttons to modify some settings?
No. Buttons are to hold shirts closed. Switches are electrical devices. Might as well start with getting the terminology straight.
I'm just asking you to let me know how would you do all of this and why.
I think you have your priorities backwards. Getting the actual temperature was not something you even mentioned. Your first question was about the packing of the finished product, which really should be the last thing you worry about.