1) Rashba allows mapping between hekdesh and monetary because] BarPada and Ulla agree what happens before the event (tree cutting); conversely, Kartabi does not map because he holds that before the event they also disagree. What's the connection?
2) Kartabi appears to making an incorrect inference in BarPada: Since pidyon is required after the event, therefore before the event, pidyon flips back to hekdesh. Why does this necessarily follow? Rather isn't the most we can say before the event is that pidyon does not work, and for all we know may have zero effect?
3) Kartabi references the Yerushalmi and states that the BarPad/Ulla debate is predicated on the Chizkiya/R'Yochanan debate. And to resolve how R' Yochanan can state that pidyon works whereas Chizkiya states that pidyon flips -- obviously referring to the case before** the event -- we say that R' Yochanan means after** the event and that they do not need pidyon. But if that's the case, then R' Yochanon and Chizkiya are not arguing about the same case!?
You have some excellent questions. Let us try to answer them.
1. Rav Moshe Kartabi says that Ula and Bar Pada have two arguments, before the cutting and after. He also says that the two arguments derive one from the other. Since Bar Pada holds that even after the cutting of the tree Kedushah cannot just dissolve, we try to understand what this person means when he says, "Ad she'Yekatzez," since even after the cutting, the tree still stays holy, so what does he mean if not that the Kedushah will "flip" back until the cutting. So the two arguments are both inserted.
2. As I already offered, the reasoning of Rav Kartabi is in the content of his words. As for the last part of your question, I tend to think that it is impossible to say Pidyon does not work. The words of a person do not create an ongoing Hekdesh, but can only create a state of Hekdesh everytime this object "loses" its Kedushah, so when this object was redeemed from the first Kedushah that originally applied, the words of the Makdish now can start the whole process again. The idea that Hekdesh can cancel the Halachah of Pidyon is unacceptable in my opinion.
3. Rav Moshe Kartabi claims that the Machlokes between Ula and Bar Pada is in both cases, before and after the cutting. He then says that Rebbi Yochanan and Chizkiyah have those two arguments too. The Yerushalmi asks a question on Rebbi Yochanan -- how can you say that even before the cutting, if the tree was redeemed it does not revert back to being Kadosh, while the Mishnah seems to say that the tree, even after the cutting, can never be redeemed? The answer is that it never needs redemption, and the Kedushah just dissolves. So Rebbi Yochanan and Chizkiyah are still mainly arguing about the status of the tree before the cutting, but the words in the Mishnah, according to Rebbi Yochanan and Ula, are referring to after the cutting.
I hope this helps a bit.
Best Regards,
Aharon Steiner