> Why the single quotes around function names?

Syntactically the same as not having them, aside from the fiddly details of string vs property name access to an object.

  thisObject = {
    usualPropertyName: value1,
    'property name with spaces': value2
  }

  thisObject.usualPropertyName
  thisObject['usualPropertyName'] // this works too
  thisObject['property name with spaces'] // you can't use the dot access style with this property name

> Why are some normal functions and others fat arrow functions?

Fat arrow functions bind "this" to whatever the surrounding context is, and if the only statement is a single object or value, it implicitly returns that.

That is to say, this:

   whatever => value

is syntactically the same as this:

   whatever => (
     value
   )

is syntactically the same as this:

  whatever => {
    return value
  }

> Why omit the parens around the param of a one parameter fat arrow function -- does that define a get accessor?

If there's only a single argument, you don't need the parentheses. This is meant mostly for the convenience of things like this:

  someArray.map(item => item.propertyName)