Answer 1:
A dangling modifier is a modifier without subject or with different befuddling subjects. Right now, 'through the report' is a modifier. It is dangling since we don't see obviously which subject (Teagan or Recommendation?) is being changed.
The best decision is the first. Flipping through the report, Teagan saw the proposal. Note that since the real subject is coming soon after the modifier isolated by a comma. It clarifies that the subject is Teagan.
Answer 2:
Here once more, the primary choice is right. While I was addressing the venture director, my sensitivities made me have a wheezing fit.
Note that in the subsequent choice, the importance of the sentence becomes as though the subject 'unfavourably susceptible response' is talking. This isn't right.