Por que Gregor "The Mountain" Clegane ainda é um cavaleiro?

In practice, the reason is of course that he was still useful to the Lannisters, who hold King's Landing and the Iron Throne. Lord Tywin might allow his faithful dog to be killed for the sake of brokering an alliance with Dorne, but certainly not for the sake of honouring a decree made by one dead man on behalf of another, neither of whom had ever been very friendly towards House Lannister.

So how did they justify their decision to keep Gregor Clegane around, given that in theory they're continuing the succession from King Robert (even though in practice House Lannister essentially seized the throne from House Baratheon)? Well, the sentence might have been pronounced in the name of King Robert, but foi anunciado by Ned Stark, a man who shortly afterwards was executed for treason. The Lannister 'party line' is that Ned betrayed Robert by plotting to take the throne away from his son and heir. Given that, why should any announcements made by Ned, even supposedly in the name of the king, be considered valid? If challenged (and who would dare to challenge them?) the Lannisters could say something like:

"Well, sure, the late Lord Stark wanted to strip Ser Gregor of his lands and titles. That was part of his plan to take the throne for himself, you see; he knew House Clegane's loyalty to the king, so from his point of view, Ser Gregor had to be eliminated so that his master plan to cheat Robert's children of the throne could go ahead."

Eddard proclaims this:

In the name of Robert of the House Baratheon, the first of his name, King of the Andals and the first men, Lord of the Seven Kingdoms and protector of the realm, I charge you to bring the King's justice to the false knight Gregor Clegane and all those who shared in his crimes. I denounce him and attaint him. (gasps, murmurs, and voices of outrage sound at court) I strip him of all ranks and titles, of all lands and holdings, and sentence him to death.

Now let's see what does that mean:

I denounce him and attaint him

Denounce means to publicly declare something/someone to be wrong or evil. Well and good, Eddard publicly declares Gregor to be evil. The second part of this statement is Attaint. Attainder is defined as

"the forfeiture of land and civil rights suffered as a consequence of a sentence of death for treason or felony."

Well and good, Eddard is taking away Clegane Keep from Gregor in name of the King and he is taking away the rights Clegane enjoyed as a landed Knight over his lands. Nothing here is mentioned about his Knighthood.

I strip him of all ranks and titles, of all lands and holdings and sentence him to death

Gregor is a landed Knight with his own lands. When Eddard declares that he is stripping him of all ranks, land and titles, he is talking about his Rank, Lands and Title as Knight of Clegane Keep. Not his personal Knighthood which he earned from Prince Rhaegar Targaryen.

For Example if a similar edict was to be issued about Lord of Tarth, that would mean he would no longer hold Tarth as his holding, He would no longer be entitled to be referred with the title of Evenstar of Tarth or Lord of Tarth, He would no longer enjoy his rank as overlord of his vassals. That does not mean he will no longer be a Knight if he is a Knight.

You cannot strip knighthood

There is a difference between Landed Knights and Knights. You can strip land from a landed Knight but you can't strip basic Knighthood from a Knight. There is simply no precedence for it.

Consider following examples:

1. Jorah Mormont: Escaped into exile for crime of selling men into slavery, stripped of his lands and titles as Lord of Bear Island, was sentenced to death but chose to flee instead. There was never an edict issued to Strip him off his Knighthood. He remains Ser Jorah Mormont.
2. Jon Connington: Stripped off his lands and titles and sentenced to exile for defeat against Robert by Aerys II Targaryen. He was no longer Lord of Griffin's Roost but he was still a Knight which is evident because he knighted Ser Rolly Duckfield.
3. Lucamore the Lusty: Broke his vows of celibacy as a King's guard and was gelded, exiled to the Wall. No one ever suggested to Strip his Knighthood.
4. Ser Jaime Lannister: Broke his vows by killing his liege King Aerys II. Punishments proposed for him included execution or to be sent to the wall however even Eddard Stark did not propose to take his Knighthood for his crimes.

Many Blackfyre loyalists fled to Essos under command of Ser Aegor Rivers after their defeat in Battle of Red grass field. They continued to be Knights despite losing their lands and titles.

There also a number of "Robber Knights" like Beric Dondarrion, The Smiling Knight etc. Despite their crimes, They are still referred to as Knights.

What is meant by Ranks and Titles

Classificação: Rank means the social standing of a person in the feudal structure e.g. Is the Person a King, A Lord Paramount, A Lord or a Landed Knight.

Título: Title means the epithets that come with the Rank e.g. King (Or His Grace), Lord Paramount of [Insert Region here] (Or his Lordship), Lord of [Insert region here] (Or his Lordship) or Knight of [Insert Tower's name here].

Neither rank or title refer to the Knighthood of the said person. If someone is a Ser, he will remain a Ser.

Why did not Lannisters carry out the former hand's orders

Porque:

1. Former Hand was a "traitor" and an enemy of House Lannister and His Grace, King Joffrey Baratheon.
2. House Clegane was loyal to Lannisters and the King and proved their loyalty with services in matter of succession and in Lannister Campaign in Riverlands.
3. Tywin was not going to give up his tool for no apparent benefit.
4. The crime, for which Gregor was sentenced by previous Hand, never occurred according to Lannisters. At least they deny they or their bannermen had anything to do with that crime, including Ser Gregor Clegane.

Lord Tywin's attitude is summarized by his famous quote:

A Task for every tool, A tool for every task

He likes to use people for whatever they are useful for. Gregor Clegane is useful for striking terror into hearts of people. In ASOS Chapter 53, Tywin explains his views about Clegane to Tyrion:

“Very well, as you ask so pleasantly. The Red Viper is not going to be pleasant, I fear... nor will he content himself with Ser Gregor’s head alone.”

“All the more reason not to give it to him.”

“Not to... ?” Tyrion was shocked. “I thought we were agreed that the woods were full of beasts.”

“Lesser beasts.” Lord Tywin’s fingers laced together under his chin. “Ser Gregor has served us well. No other knight in the realm inspires such terror in our enemies.”

So, after the Downfall of Eddard Stark, Lannisters took control of office of the Hand and regency.

Lord Tywin was well disposed to get Gregor killed if there was some incentive for him in it e.g. Winning Dornish friendship and end of the blood feud between Lannisters and Martells. But Killing Clegane just because a traitor (Eddard Stark) had commanded to, doesn't serve Tywin at all and leaves him bereft of his most useful monster who had served very well for Lannister Cause.

Eddard condemned Gregor to death for crime of massacring Riverlanders in raids. Lord Tywin ordered those raids and was careful enough to make the raiders operate without any banners or evidence that could link the raids back to Casterly Rock. The raiders were Common Outlaws. So why should Tywin carry out sentence for a crime which Gregor never did and was falsely accused of according to Tywin's position?

Remember, punishing Gregor for raids on Riverlands directly incriminates Tywin Lannister as well.

What became of Eddard's sentence?

It would presumably be considered null and void given that the man was shortly convicted of treason. There is no explicit mention of Tywin or Tyrion as Hands trying to de-jure nullify the order. Which probably means that in Westeros, an order being de-facto null and void is good enough.

In the books, Tyrion organizes a treaty with the Martells, and one of the conditions is that the murderer of Princess Elia will be brought to justice. It is implicit in Tyrion's understanding that this means that the Mountain will be brought to justice and executed, and in Uma tempestade de espadas, Oberyn Martell arrives in King's Landing seeking that justice. However, Tywin Lannister tells Tyrion that the Mountain is too useful to them to give up to the Martells. I won't spoil too many plot points.