I think Eliezer believes that chickens aren't sentient, and at the time of writing HPMOR, he probably thought this was the most common position among people in general (which was later contradicted by a poll he ran, see https://www.facebook.com/yudkowsky/posts/10152862367144228 ). If Dumbledore believed that chickens weren't sentient, he might not think there's anything wrong with setting fire to one.

For lots of discussion about Eliezer's and others' philosophy of mind, see https://rationalconspiracy.com/2015/12/16/a-debate-on-animal-consciousness/

The animal that appears to be a chicken to Harry isn't a chicken. It's also not simply another creature transfigured into a chicken. It's a mystical creature called a phoenix and part of the inherent nature of what it means to be a phoenix is to burn and then rise from the ashes.

Yudkowsky plays with a reference to the tastes like chicken meme.

I'll have to go back and re-read - was it clear that the chicken that burned wasn't actually Fawkes? I took that scene as Harry's interpretation of "normal" phoenix renewal.

As to your questions, I believe the standard non-magical answers apply pretty well:

1. Almost nobody opposes the creation of animals (or people) by any possible means (today that's breeding or cloning), even though they're expected to fade. Why oppose it here?

2. Why is it wrong to burn a real chicken alive? If I thought there was an important lesson to teach a human, I'd do that in a heartbeat. It's a chicken, it has very low moral weight to most people. In fact, I burn chicken often, then eat it (granted, I have someone else kill it and dissect it first, but that's not an important moral distinction IMO).

I'm not sure how magically plausible this is, but Dumbledore could have simplified the chicken brain dramatically. (See the recent SSC posts for how the number of neurons of an animal correlates with our sense of its moral worth.) Given that the chicken doesn't need to eat, reproduce, or anything else besides stand and squawk, this seems physically possible. It would be ridiculously difficult without magic, but wizards regularly shrink their brains down to animal size, so apparently magic is an expert neuroscientist. If this was done, the chicken would have almost no moral worth, so it would be permissible to create and torture it.