This dof was very interesting, but very frustrating. that is, how could the entire gemara
not answer the questions.
My questions, then, are two:
1) What, exactly is the nature of a teiku? What strength does it
have in relation to a sugiah that seems to answer it up? What are
the ramifications for all these inyonim?
2) How are each of these inyonim leading to a teiku?
1) A "Teiku" means that the Gemara remains with a doubt concerning the original question, and the question is not answered from anywhere else or any other Sugya. (If indeed we find another Sugya that takes for granted one of the two sides of the doubt of the Teiku, then it must be that that Sugya argues with the Sugya that leaves it as a Teiku. Regarding the Halachah, Tosfos (Avodah Zarah 6a, DH Iba'is Eima) writes that the Halachah follows the Sugya that answered the doubt.) Regarding the Halachah when the Gemara leaves the question as a Teiku (and there is no other Sugya which argues and answers the doubt), there are various rules of P'sak in cases of Safek that apply (sometimes the Halachah l'Ma'aseh will be l'Chumra, and sometimes l'Kula, for example).
2) Since we do not find any proof -- from a Mishnah nor from a Beraisa -- to either side of the question, it remains a Teiku. (Regarding some of the questions in our Sugya, some Rishonim explain that they were asked in the form of "Im Timtza Lomar" (that is, the present question is based on the assumption that the answer to the previous question is known to be one of the two sides), and some Rishonim say that every time there is a question which is an "Im Timtza Lomar," it means that the answer to the previous question is exactly as expressed by the "Im Timtza Loma." The second question is being asked only because there was an answer, from logic, for the first question.