Does Needletail deserve her name?

No, she doesn't deserve a warrior name. In fact, I'm surprised she even chose one, considering how much she detests the Clans. It would make more sense to go by Needle.

Now that you mention it, I think that speaks a bit about then workings of Needletail's mind.
Perhaps she's far more attached to the clan traditions that she's letting on, thus continuing with the idea of a warrior name. 

Or maybe she wanted a warrior name b/c that's all she knows. Idk. I'm 80% sure she's a sociopath.

Now that is the best name for her 100000000000000/1

Should've changed her name to Needle.

But if she goes back to the Clans she doesn't deserve to be a warrior.

This.

Agreed.
The clans had been enjoying many seasons of peace, and these young apprentices didn't have to face the difficulties that the older warriors struggled through to keep everyone alive.
And because they have no respect for their elders, and have no interest in spending time with them, they're not able to learn about past events and, thus, grow from them. As such, their knowledge and mindset is limited to the few moons of experience that they've accumulated.
Ironically, though they claim to have open minds and criticize the others for not being like them, they're very narrow-minded as all they see and care about is what has occurred within their lifetime, instead of trying learn from the past and attempting to avoid repeating it.
They select the very irrelevant portions of Shadowclan history, like being intimidating, instead of looking at all the leadership challenges, threats, wars, battles, ect. that Shadowclan had to face to get to where they are. They claim that they want to return everything to how it was, but they don't realize what they're asking for. 
It's like if teenagers now a days were arguing that we need to return to the 1920s and 1930s way of life and government, not realizing that that exact lifestyle is what lead to the Great Depression, which is something we want to go out of our way to avoid repeating.