Answer:
1. The author's arguments that supports the closing statement "That said, the common claims made against veganism are false" and would be most convincing to someone that opposes veganism would be the point made about animals not being worthy of certain legal protection.
The author made an argument against that claim by referencing the work of the Australian professor of bioethics Peter Singer who made a case for the protection of animal rights based on the idea of utilitarianism which states that pain more than intelligence is the most important determiner of rights.
2. The author begins to look at counterarguments to veganism in the second paragraph after clearly describing how non-vegans see the actions of vegans as absurd and unbearable in the first paragraph.
I think she chooses to start her counterarguments because she wanted to prove wrong some of the claims by "meat eaters" that vegans have horrific habits and she also wanted to use it to restore "balance and maturity".