Word types + に/を + する

私は出来るだけ静かにしていた。 I kept as quiet as possible.

The する, used with the continuative form (連用形) of an i-adjective or na-adjective, or an adverbial phrase such as こう、そう etc., is an intransitive verb meaning そのような行動や態度をとる ("to behave/act in a particular way").

明鏡国語辞典 states:

する ?〘自動詞〙
❻《様態を表す形容詞・形容動詞の連用形、「こう」「そう」など、副詞句に直接付いて》 そのような行動や態度をとる。「静かにしろ」「人に優しくする」「彼女とは常日ごろ親しくしている」「言われたとおりにする」「こうすればいいでしょう」

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1. Does 静かにして more literally mean "made quiet", how does it differ from になる?

No, 静かにする literally means "to behave/act quiet" → "to keep quiet".
「神妙にする」 "to be docile"
「勝手にする」「好きにする」 "to act as one pleases"
静かになる means "to become quiet (after not being quiet)". It involves a change of state or transition from being not quiet to being quiet.

2. Does this syntax of にする work the same for i-adjectives (面白いにする)?

You can use this する with i-adjectives, but not 面白いに. The continuative form of i-adjectives ends with ～く. eg:
優しい → 優しく 「人に優しくする」 "to be kind to others"
おとなしい → おとなしく 「おとなしくする」 "to stay quiet"
よい → よく 「友達と仲よくする」 "to get along with friends"

1. What is the difference between using を and に?

(I don't really know what you're asking but) ～をする can mean many different things. Used with an を, the する is transitive. eg:
勉強をする to do studying → study
教師をする to work as/play the role of a teacher
ネクタイをする to wear a tie
etc...

And ～を‥にする means "make/turn ～ (into)‥" or "choose ～ as ‥". eg:
牛乳をチーズにする "make milk into cheese"
部屋をきれいにする "make a room clean"
山田さんを議長にする "make Yamada-san a chairperson / elect Yamada-san chairperson"
ワインをお土産にする "choose wine as a gift"
etc...

2. Is this grammar point used only because the noun isn't a suru-verb, for example does 勉強に/をする not work?

Well, it's because the する has a different meaning.
You can say "Noun+に+する" using this する, though it doesn't seem productive, since this is the only example that I can think of:
いい子にする lit. "to behave/act a good child" → "to be a good boy/girl".

勉強にする doesn't make sense.

Adjectives are words that modify nouns, whereas adverbs are words that modify adjectives and verbs.

By adding に to a na-adjective, you turn it into an adverb. In this way, 静かにする can be seen as the adverb "quietly" directly modifying the verb する "to do":

静かにする。Literally "to do quietly", i.e. "to be quiet".

I think that this sense of "to be quiet" fits nicely in your example sentence. Also note that 静かにする is a very common sentence so it can be thought of as a set phrase.

However, note that you could use 静かに as an adverb to modify other verbs, too:

静かに寝ている。Sleeping peacefully/quietly

On the other hand, to turn an i-adjective into an adverb, you cannot attach に. Instead, drop the い and replace it by a く. For example:

速い fast (adjective) → 速く fast (adverb), rapidly

あの車は速く走ります。That car runs fast.