異音の原因がわかり、安堵して大きく息を吐く
Here's my thoughts, can someone tell me whether I'm right or not?
realise cause of the sound > therefore feels relieved > then they sigh.
I thought both these usages of the continuative form were both the usage of 原因・理由を表す
but I'm not quite sure about the literal meaning of for the last part.
My main question is whether it expresses causation or merely states the actions in which they happen.
Your guess is perfectly right. [EDIT for the update: the basic meaning of continuative forms is sequential occurrence, may or may not imply causality.]
And the last part 大きく息を吐く describes a motion of deep breath (expiration). I think English speakers give "sigh of relief", too. Perhaps it's more understandable in English to merge the last te-form verb chain into one action, as:
I breathe a deep sigh of relief, after realizing the cause of the strange noise.
It's a common gesture to exhale deeply when you're relieved, as the word ほっとする "feel relieved" comes from the breathing sound ほっ. Other idiomatic expressions of relief include 胸{むね}をなでおろす (lit. "stroke down one's chest") and 一息{ひといき}つく (lit. "take a breath").
