So...to clarify this in my own mind, 'de nouveau' means 'again'? What about 'il casse encore'? Or does this sentence 'il casse de nouveau' more accurately mean it is newly broken?
it is an idiom. we have that in port and span too: de novo, de nuevo. roughly ir means freshly broken.
Yes and it is the same for all verbs of the 1st group (all finishing by -er at the infinitive form), like manger, danser, parler, etc...
If 'it' was femine, would this be 'Elle casse de nouvelle'? If 'it' was plural, would it be 'Ils cassent de nouveaux'? Or is 'de nouveau' an adverb that doesn't change?
"de nouveau" or "à nouveau" (both correct in this sentence) are adverbs in two words, which means that "nouveau" will remain invariable, as if you had used "encore".