Both are acceptable. You are right that "were" would be more grammatically correct, but in everyday speech people frequently use "was" nowadays.

That is a good question. There are different ways to say the same thing. "She felt as though she were a bird..." would be correct. However, there is another problem with the sentence. Have you learned about dangling modifiers yet? "at this moment" is describing how she feels, not the bird, so "at this moment" should be next to "she felt". The two following two options are the best, but all combinations can be heard in casual speech. "At that moment she felt as though she were a bird." or "She felt at that moment as though she were a bird." Personally, I would say "At that moment she felt like a bird."

deletion

if you want to use "were" instead of was, here is an example - "if i were a bird, i would fly in the sky".

i think it is correct "She felt like that she was a bird at that moment"