The translation doesn't mention anything that refers to する.
Does 青々した芝生 suggest the lawn being made green or lush?
する doesn't always mean "to do" or "to make". 青々(と)した expresses a state here, meaning "(fresh) green", as in the given translation.
I suppose 青々した芝生 is the exactly same as 青々とした芝生.
It seems to me you said と is always optional without changing the nuance or meaning.
It sounds like that is not only optional but redundant.
Both are equally used, thus, it's not "redundant" at all. This is a fact whether you feel it's redundant or not. Sorry if this sounds harsh, but you seem to be baselessly confident with your own impression. Actually, it seems to me that you are just inexperienced.
What could be difference if one says 青々芝生 or 青い芝生?
青々芝生 is ungrammatical. 青々 is an adverb, as we already pointed out, thus, it can't modify a noun directly as same as もっと can't modify 人 or 水.
青い芝生 is valid. Have you never heard a proverb 隣の芝生は青い, which is the Japanese translation of "The grass is always greener on the other side of fence."?
It seems to me 青い tends to mean blue rather than green. The dictionaries I use say the same thing.
青 has a nuance of young, inexperienced, immature (e.g. 青二才, 青くさい, 青葉, 青年), similar to "green" in English. It doesn't always express only the color of the thing also in Japanese.
It seems to me where the adjective of single word repeating itself such as 青 -> 青々 the result becomes an adverb. Perhaps it might not always be true.
There are many those examples, and also many counterexamples.
e.g.
〇熱々 あつあつ
△冷々(ひえびえ exists, but it's usually written as 冷え冷え or ひえびえ)
〇寒々 さむざむ
×暑々
〇深々 ふかぶか
△浅々(obsolete)
〇近々 ちかぢか
△遠々(遠々し is obsolete, and 遠々 is not an adverb.)
〇早々 はやばや/そうそう
△遅々(The reading is ちち, not おそおそ)