で used to indicate the state of something

Question

I came across this usage of で in this article.

彼はお酒を熱燗で飲む
He drinks sake in a hot state.

Is this で the て form of the copula or the particle で?

• If its the て form of the copula, why is を and not が? What is being coupled? Is it grammatical to basically have the main clause chopped in the middle?

• If it's the particle で, how could it be that this 熱燗で tells something about the object as opposed to telling something about the verb (i.e. the "limit" within which it applies).

Answer 1

It's the particle で, definition #4 of 格助詞「で」 in 明鏡国語辞典:

で
🈩〘格助〙
❹ 動作を行うときの様態を表す。
「急ぎ足で歩く」
「親子で出席する」
「笑顔で答える」
「土足で上がる」

It expresses [様態]{ようたい}, in what manner some action is performed.

It's not the copula だ→で since the sentence cannot be split to ～～熱燗だ+飲む.

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how could it be that this 熱燗で tells something about the object as opposed to telling something about the verb

I see it as more like 熱燗で飲む (how you drink it)...

「ポン酢でいただきます」「[生]{なま}で食べる」などの「で」だと思います。「お肉を低温で調理する」「ビデオを２倍速で再生する」なども同じ「で」だと思いますがどうでしょう。