Question 1: Are these two things about right? I assume that using うすい changes little in the meaning. What I gather is that the わたし is talking about the 商人's fragility.
I can't be completely sure without more context, but the first assumption is incorrect, and therefore the second point too.
This particular usage is commonly seen in mystery stories or in a similar setting. dictjuggler's translation corpus has this example:
共犯者の線も濃厚だ: evidence of an accomplice is definitely there
My understanding of this collocation is:
線 line of reasoning
The closest sense listed in Daijisen for 線 is "6. 物事を行う道筋・方針" (policy, principle), and for うすい, "3-5. 可能性があまりない" (slim possibility).
And also, there's another reason why I said it's unlikely that the merchant is the culprit.
[culprit] ------> [merchant]
(I'm wildly guessing here that there's some kind of crime involved.)
Question 2: Is there a clear etymology here?
I'm not sure how or why this usage came to be. You could get away with exchanging 線がうすい for 線がほそい behind the reader's back, and your intention will eventually get through, but they'll either do a double take, or just unconsciously replace it with the more natural collocation.