PA: First off, your tone is refreshing and I very much appreciate it. You raise valid points -- ones I've considered myself -- and have honed in on an important passage for this discussion. I'll try to sum up my understanding of the passage here, but fleshes things out more thoroughly. I've used it to help gather my thoughts here, although my opinion was informed by a variety of sources. A good book on this topic is by Richard Clark Kroeger and Catherine Clark Kroeger; it elaborates well on the pagan and gnostic beliefs relevant to the false teachings in Timothy's church.
"Have/Usurp Authority"
The Greek word authentein is the word which is translated "exercise authority" in the version you quoted. It is also sometimes translated "domineer" or "usurp authority." The Vulgate, Geneva Bible, and the King James are examples of translations that use this latter sense of the word, implying a negative, "taking-control" kind of action.
The reason why there is some ambiguity about the meaning of authentein is that it only occurs once -- here -- in the New Testament. So one of the most useful tools of translation (looking at the word in another scriptural context) isn't available. Therefore, one turns to other Greek texts in order to figure out the nuances of the word. Ancient texts prior to the writing of the New Testament most often do not use the word in a neutral sense of "have authority," but use it to refer to domination or crime -- even murder. It carries with it strongly the notion of one grabbing power over another, often unjustly. In a much later period, this word came to be used in the more neutral sense, but this was significantly after the New Testament was written, and is therefore given less weight than the usage which was in place in the earlier period.
Paul's choice of authentein is interesting, since he does not use it elsewhere in his known writings. There are other Greek words which he could have used to denote simple authority without negative connotations. Proistemi is a word for leadership which he uses several times and would have been natural to use here if Paul were concerned solely about ordinary standards for leadership.
Therefore, many scholars conclude that there are contextual concerns which informs Paul's choice of word here. Paul knew the specific concerns of the Ephesian church and the difficulties which Timothy would have to address there; given indications that the church was experiencing a particular problem with women, these scholars believe the evidence points to a specific, localized application of this principle, not a broad, universal one.
Adam Was Created First
If I understand you correctly, the aspect of this passage which you find most compelling is Paul's use of the creation account to defend the position. This does give one pause, certainly -- who would want to go against the created order? However, once again there are contextual issues not readily apparent to us, but which are rendered no less legitimate for the fact that it requires study to learn them.
John Davis, the author of the article linked above, does a good job of laying out examples of Paul's use of the creation account in various epistles to make various points. Paul drew imagery and analogy from Genesis to form particular applications in particular congregations. At times, he uses Genesis differently than at other times -- but that is not contradictory, as he is speaking to different situations which require a different pastoral response. This was a frequently-employed rhetorical method of the time, although it isn't as common today.
Distinct circumstances existed in Ephesus which makes it extremely likely that Paul was particularly concerned with false teaching coming from the women of that community, relating to women in general. The book cited above gives a more detailed explanation, but I'll sketch the outlines.
Ephesus was known for its temple to Artemis, who among other things was celebrated as the goddess of birth. In that region, there was a strong emphasis on motherhood as the source/originator of life. There was a proto-gnostic teaching there that woman was the originator of man; therefore, some scholars (the Kroegers, for example) think that the prohibited authentein may refer to women teaching that they are the originator of man.
As these pagan beliefs blended with bits of Christianity to form the beginnings of gnosticism, the idea emerged that Eve pre-existed Adam, and that he came into existence through her. This led to a multitude of false teachings, because it privileged one gender (female) over the other (male). Women influenced by such thought must have created a disturbance when they tried to share such teachings in Timothy's church. Therefore, it was important to correct this idea that that the woman came first. Paul's words are a sort of "putting women in their place," but only inasmuch as these particular women were trying to elevate themselves above their brothers. Read with knowledge of ancient Ephesus and gnostic beliefs -- namely, that Eve was created first, and women therefore had a special power which men lacked -- Paul's words become a very pointed rebuke to a specific false teaching.
Other Scriptures
This post has already gotten pretty long, so I'll just make a brief mention of one other thing which informs my interpretation of this passage: Other parts of the Bible. I believe, as I'm sure you do, too, that scripture will not contradict itself. Therefore, if something seems contradictory, then I look deeper to see if there are any contextual issues which explain that. As I see evidence in scripture of God blessing women in authority (Deborah, Huldah, Phoebe, Priscilla, Junia, and Nympha are the ones who first spring to mind), I believe that God would not elsewhere state an absolute principle which would condemn the work of these women. That means that the contextual issues I've mentioned are especially valuable as I try to understand the whole of scriptural teaching regarding gender roles and women in leadership.
I hope this has been helpful... as long as this post is, still I feel like I've only scratched the surface. I hope this helps you (and anyone else who has read all the way through it) to understand why someone might consider my position to be a legitimate and biblical one. Thank you for taking the time to read and consider these points.