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Rabbi Aqiva Eiger (shu”t #136) says that there are two types of doubt, and each has its own mechanism for birur, for clarification. The case of qavu’ah is one where the reality was once established. So in principle, there is a specific halakhah assigned already to this case. The doubt is in what that halakhah is. In this situation, we do not invoke rules ...
The proof essentially functions as a process of elimination: by showing that the alternate explanation is not viable, you demonstrate the validity of the acceptable explanation. Also, if the gemara only gave the correct answer, the reader would never learn how to analyze and reason on his own. "Give a man a fish and you feed him for a day; teach a man to ...
it would be interesting according to R Yosef engel ( Lekach Tov, Klall 1) who entertains a third option as to how shlichus works, in that the shliach remains an independent entity, while his creation of a specific halachic chalos works and is effective for the meshaleach, where as the first mehalech is power of attorney, and special strength is invested in ...
As @Fred basically notes in the comment, the "davar" in the pasuk which requires "al pi shnayim" is the actual event/act itself, not the myriad details and preconditions that enable it (e.g. soklin al hachazaka by incest). Further, it would be hard to apply an anan sahadi to a scenario that does not conform to normal human nature, such as murder.
Rashash (on the mishna in the g'mara) cites a machlokes as to who has to provide the location of the divider: Rashbam says it's the seller [as do Rabenu Ger'shom on the page of the g'mara, alleged Rashi on the page of the Rif, and Nimuke Yosef], while Rambam (commentary on the mishna) says it's the buyer.
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