Evolution question - asian eye skin flap

I don’t see an animal capable of making that kind of distinction. It would seem to me that if a female of the species decides that a trait is attractive and therefore wants to mate with that male, THAT is what will help a trait spread. But I don’t see how an animal could say “that tail is ridiculous, but he must be fit to drag it around.”

Do animals have that capability?

They don’t make that distinction consciously. Here’s the scenario I imagine:
Before the showy tail feathers evolved, males had just…tail feathers. But males that were diseased, or starving, or had some hormonal imbalance had bedraggled looking feathers. Females then evolved an aversion to nasty feathers as a way of avoiding bad mates. This was just a simple mental trigger: smooth, long, colored = sexy. Once this trigger was in place in the females, it provided a selective pressure in the males to over-stimulate the trigger by producing longer, more colorful tail feathers. Now the pressure was on the females to be more selective in their triggers. Iterate.

No, but they don’t have to. The actual trigger for a female to find a male attractive may be the size and color of the tail. The handicap principle is just an explanation for why an attractive tail may be what is known as an “honest signal” of the physical fitness of the male.

Absolutely - a large part of “attractiveness” is a judgement of health and fitness, including the ability to raise young or protect a family. It may not be a conscious reasoned thing, but it’s part of what makes a potential mate “attractive.”