My dad is petitioning for his brother (my uncle) and by default, his family (my cousins & aunt). My cousin is over 21 as a result of aging out, but the petition was filed before she turned 21. Next year, she has to file as an independent on her tax return, because she no longer qualifies for dependent child/relative on my uncle's return. Will this affect her status as a child of my uncle for the petition purposes? Will she still be able to obtain a permanent resident card/apply for green card when/if the petition for my uncle gets approved?

No. Whether someone is a dependent for taxes has nothing to do with being a child for immigration purposes. The latter just has to do with age.

Note that just because the petition was filed before she was 21 does not mean she does not age out. The time the petition was pending will be subtracted from her age. But the time the petition was pending will still be many years shorter than the wait time for a visa number to become available for the F4 category, which is currently almost 13 years for most countries.

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... My cousin is over 21 as a result of aging out ...

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What does "aging out" mean? In other words, she is over the age of 21? If that's what you mean, the "aging out" part is redundant.