How well does the Wall work?

Question

The Wall was built to keep out White Walkers (or The Others, if you prefer).

Have there been any "long winters" ever since its construction? How do we know that it serves its purpose, i.e. to repel Others/White Walkers?

(I do not mind spoilers.)

Answer 1

We know that (spoiler ASOS/season 5?)

Coldhands

who is a wight that it is claimed cannot pass the wall. We also know that (spoiler ADWD/season 4)

Melisandre

can sense that the Wall has great power.

There is also a great comparison implied between the two constructions when we hear Melisandre talk about the magical fortifications of Storm's End, when Davos smuggles her beneath the walls to send her shadow baby to murder Ser Cortnay Penrose. We know from various sources that Brandon the Builder built both the Wall, and Storm's End. Melisandre's difficulties at Storm's End hence also implies that the Wall has similar powers.

And we have heard from Mance Rayder that he believes that the Wall will protect them: He has not come as a conqueror, he has come to hide behind their wall.

But as of yet, we do not really have any direct evidence of the Wall repelling White Walkers or wights, besides the above one. There has only been one Long Winter before: The Long Night.

Answer 2

80% Well

As far as I know it has never been tested against the actual White Walkers themselves since it's construction. However, we do get a few hints as to the Wall actually protecting against different threats. For starters it's really high, 700 feet apparently.

The Wall is three hundred miles long and seven hundred feet high, Sam reminded himself. If they kept going south, they had to find it, sooner or later.
_{Storm of Swords, Samwell III}

Such a large Wall is already great protection against normal threats, such as the wildlings, as it is already difficult to pass.

The Wall also contains The Black Gate that can only be opened by a man of the Night's Watch and can also apparently only be found by them.

"There's a gate," said fat Sam. "A hidden gate, as old as the Wall itself. The Black Gate, he called it."
The Reeds exchanged a look. "We'll find this gate at the bottom of the well?" asked Jojen.
Sam shook his head. "You won't. I have to take you."
"Why?" Meera demanded. "If there's a gate . . ."
"You won't find it. If you did it wouldn't open. Not for you. It's the Black Gate." Sam plucked at the faded black wool of his sleeve. "Only a man of the Night's Watch can open it, he said. A Sworn Brother who has said his words."
_{A Storm of Swords, Bran IV}

Sam seems to think it might also hide itself, whether he means from a none sworn brother or because it wants to is up for debate.

"Demons made of snow and ice and cold," said Stannis Baratheon. "The ancient enemy. The only enemy that matters." He considered Sam again. "I am told that you and this wildling girl passed beneath the Wall, through some magic gate."
"The B-black Gate," Sam stammered. "Below the Nightfort."
"The Nightfort is the largest and oldest of the castles on the Wall," the king said. "That is where I intend to make my seat, whilst I fight this war. You will show me this gate."
"I," said Sam, "I w-will, if . . ." If it is still there. If it will open for a man not of the black. If . . .
_{A Storm of Swords, Samwell V}

The Wall itself is also more than just Ice and contains some magic/spells that can stop the dead/wights from passing.

"He wasn't a green man. He wore blacks, like a brother of the Watch, but he was pale as a wight, with hands so cold that at first I was afraid. The wights have blue eyes, though, and they don't have tongues, or they've forgotten how to use them." The fat man turned to Jojen. "He'll be waiting. We should go. Do you have anything warmer to wear? The Black Gate is cold, and the other side of the Wall is even colder. You—"
"Why didn't he come with you?" Meera gestured toward Gilly and her babe. "They came with you, why not him? Why didn't you bring him through this Black Gate too?"
"He . . . he can't."
"Why not?"
"The Wall. The Wall is more than just ice and stone, he said. There are spells woven into it . . . old ones, and strong. He cannot pass beyond the Wall."
It grew very quiet in the castle kitchen then. Bran could hear the soft crackle of the flames, the wind stirring the leaves in the night, the creak of the skinny weirwood reaching for the moon. Beyond the gates the monsters live, and the giants and the ghouls, he remembered Old Nan saying, but they cannot pass so long as the Wall stands strong. So go to sleep, my little Brandon, my baby boy. You needn't fear. There are no monsters here.
_{A Storm of Swords, Bran IV}

Note that this passage also states "so long as the Wall stands strong" so breaking the Wall could also break the spells and magic inside of it.

Note that Coldhands/Benjen Stark is a wight saved from turning to a White Walker wight by the Children of the Forest.

Benjen Stark: I led a ranging party deep into the North to find White Walkers. They found us.** A white walker stabbed me in the gut with a sword of ice. Left me there to die. To turn. The Children found me. Stopped the walker’s magic from taking hold.**
Bran Stark: How?
Benjen Stark: The same way they made the Walkers in the first place. You saw it yourself.
Bran Stark: Dragonglass. A shard of dragonglass plunged into your heart.
_{Game of Thrones, Season 6 Episode 6, "Blood of My Blood"}

And we know that Benjen must be dead because of what he tells us himself.

Benjen Stark: The Wall is not just ice and stone. Ancient spells were carved into its foundations. Strong magic to protect men from what lies beyond. And while it stands, the dead cannot pass. I cannot pass.
_{Game of Thrones, Season 6 Episode 10, "Winds of Winter"}

Note again the comment of "while it stands".

As such we know that the dead cannot pass through the Wall and so it has defence against the White Walkers' army, their wights. However, we do not know if it can stop the White Walkers themselves as they are most likely living creatures.

Melisandre smiled. "Necromancy animates these wights, yet they are still only dead flesh. Steel and fire will serve for them. The ones you call the Others are something more."
"Demons made of snow and ice and cold," said Stannis Baratheon. "The ancient enemy. The only enemy that matters." He considered Sam again. "I am told that you and this wildling girl passed beneath the Wall, through some magic gate."
_{A Storm of Swords, Samwell V}

The only time the Wall has been tested was in Season 7 Episode 7 of the show and in that case the following happened.

The Night King rode on the wight/White Flyer Vicerion and used his Icey Fire breath to destroy the Wall, the dead and White Walkers could then pass. Note that the Wall is now not strong or standing and so this brings us back to the previous comments. You can view the scene here: Season 7 Episode 7, The Night King and Vicerion destory the Wall, youtube

In conclusion the Wall works well but it has never truly been tested against the White Walkers alone.

---

As for your secondary question "Have there been any "long winters" ever since its construction?" I assume you want to know if it could withstand a long Winter. Well the Wall is made of Ice so a long Winter would probably do it more good then bad. As for actual "long winters" since it's construction the following wiki page only states the longest known winter to have been 5 years long. All dates are given in AC and not BC so it is possible that there have been longer winters BC.