後には、その基準を流れのあるものととらえ、一緒に進む一続きの行動を表す文が来る。(part of explanation for ～に沿って)
俺のとまどいを不安の表れと思ったのか、そんなふうに声をかけて、彼女は教室を出て行った。
I've treated the を in these sentences as の. How is を used here, and what is the best way to interpret it? I assume it has something to do with the と-perception verb following it, but it doesn't line up for me.
I think you need to look at parsing the sentences in a different way, since you are cutting off the scope of your analysis too soon. I think the real pattern here is "N を N と V", where V is a verb of perception. Both of these sentences use that pattern to express perceiving one thing as another thing.
...
For the first sentence, I would split it up as follows:
［その基準］を
［流れのあるもの］と
とらえ
The verb とらえる means something like "to perceive". What is being perceived is "その基準", and what it is being perceived as is "流れのあるもの".
...
For the second sentence, I would split it up as follows:
［俺のとまどい］を
［不安の表れ］と
思った
The verb 思う means "to think". What is being thought about is "俺のとまどい", and what it is being thought as is "不安の表れ".
とらえる is a verb that requires を to indicate the direct object
Example: 魚をとらえる (to catch a fish)
https://dictionary.goo.ne.jp/jn/160699/meaning/m0u/捕える/
One of the meanings of とらえる is "to understand", "to interpret":
物事の本質・内容などを理解して自分のものとする。把握する。
と is particle that can be translated as "as".
So the translation becomes:
その基準を(this standard)
流れのあるものと(something with a flow)
とらえる(understand)
その基準を流れのあるものととらえる
To understand this standard as something with a flow.
