規則を破った生徒は 厳重に処分する.

厳重 is a NA adjective meaning strict, but why is NI used here? Why isn't NA used?
I checked the dictionary of japanese grammar and i'm sure how to classify this NI.
Its not a location, time, etc.
It seems more like a manner/method of doing something..if so, can DE also be used here?

に makes it into an adverb.

Adverbs are regularly made from adjectives by using ni after na-adjectives and conjugating i-adjectives to the ku-stem, so,

静かに　＝　quietly

早く　＝　quickly

These two are very common and you will often hear them used on their own in anime etc. A teacher may say 静かに (quietly=be quiet) to a class, and people will often say 早く! (quickly!=hurry!).

This abbreviated usage probably derives from the fact that once an adjective has been made into an adverb it can then become a quasi-verb by adding suru.

静かにする　＝　"do quietly" (or perhaps more accurately, "act quietly"), ie be quiet.

Like so much of Japanese this is really very simple and regular and can be used to do all kinds of things.