1) I think it is adverbial. 2) The second one is just as adverbial as the first. 3) I think your versions are fine. I don't see that improvement is really needed.
My paraphrases would be:
1) Now that I've read the job description, ... 2) Once I read the job description, ...
I just thought my first sentence doesn't really show the relationship between the ing clause and the main clause. The ing clause doesn't answer where when why etc, whereas...
in my second sentence it does: how did you realise? By having read.
Your paraphrases also don't make a connection, just as my first one doesn't.