Does the structure in bold mean: "I have immediately tried to touch the seat"?
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I would not translate the structure in bold (触ってみたら) that way.
First off たら is indeed a conditional and it's key to understanding the construction as a whole.
The latter part means "it was cold enough to make me shiver."
The left hand part means "when i tried to touch the seat"
*シート = either seat or sheet * 触る (here conjugated as 触って) = to touch * ~てみる is a helping verb that means to "try"
I'd actually say とっさに = for a brief moment applies to the entire construction.
So then we get "when I tried to touch the sheet/seat for a moment it was cold enough to make me shiver"