TL; DR

The Short Version

There were two agreements:

1. Richard's lands were under the protection of the church until he returned from Crusade. (Standard operating procedure while kings and lords were on Crusade)
2. Richard got to keep the lands which he had received in Normandy as dowry for Philip's sister Alice (or Alys), after he broke the betrothal, in return for a payment of 10,000 marks. (agreed between Richard and Philip in Sicily en-route to the Crusade)

Yes, Philip broke both agreements.

No, he didn't manage to conquer the Duchy (at least, not until after Richard's death).

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The Longer Version

(although still very much a summary!)

The first point to remember in all this is that while he was on crusade, all of Richard I's lands were supposed to be under the protection of the church (as, indeed, Philip's lands had also been, while he was on crusade). In theory, Philip II was unable to attack them until Richard returned to his own lands.

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The Third Crusade

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The Third Crusade was a response to Saladin's capture of Jerusalem. King Henry II of England and King Philip II of France ended their endemic conflict with each other in early 1189 in order to jointly lead a new crusade to the Holy Land with the goal of recapturing Jerusalem. Henry II died on 6 July 1189, and the command of the English contingent passed to his successor Richard I. As part of the peace settlement between England and France, it had been agreed that Richard would marry Philip’s sister Alice (or Alys) on his return from the coming crusade.

In late 1190, both kings had arrived on Sicily, en-route for the Holy Land. Relations between the two men were always fractious, and the situation was not improved when, early in 1191, Richard broke off his long-standing betrothal to Alice and announced that he would marry Berengaria of Navarre. Richard made matters worse by claiming that Alice had been Henry II's mistress, and that she had borne him an illegitimate son.

To safeguard the progress of the crusade (or, at least, to ensure that he could not be blamed for its failure), Philip agreed to accept a 10,000 mark payment from Richard in lieu of Alice's dowry. This dowry had included the lands of Vexin, and the strategic fortress at Gisors. I believe that this is probably the agreement between Philip and Richard that you referred to in the question.

The crusade itself didn't go particularly well for Philip. He contracted dysentery during the siege of Acre which seriously compromised his health. He failed to achieve the military glory which he had sought, as his exploits during the siege were easily outshone by those of King Richard. He also lost a key ally in the person of Philip, Count of Flanders, who had died during the siege.

This last was also politically significant since it threatened to undermine the Treaty of Gisors, and Philip used this as a reason/excuse to return to France in order to settle the succession in Flanders. Obviously, this meant breaking the vow that he had taken when he took up the cross, and Richard was characteristically blunt about it:

"It is a shame and a disgrace on my lord if he goes away without having finished the business that brought him hither. But still, if he finds himself in bad health, or is afraid lest he should die here, his will be done."

Philip left Acre and returned to France via Genoa in August 1192.

Richard continued with the crusade, eventually coming to terms with Saladin and signing a treaty in September 1192. Richard left the Holy Land a month later.

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Richard in Captivity

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While en-route to England, Richard was captured and imprisoned by Leopold V, Duke of Austria. Leopold was another noble who had fallen out with Richard following the siege of Acre, and departed from the crusade early. Richard was subsequently transferred to the custody of Henry VI, Holy Roman Emperor. It took a "king's ransom" of 150,000 marks to obtain his release, and he didn't get back to England until March 1194.

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In the meantime, Philip had been busy. In December 1192, Henry VI and written to Philip, stating:

"... we have thought it proper to inform you of what happened to Richard, king of England, the enemy of our empire and the disturber of your kingdom, as he was crossing the sea on his way back to his dominions.

… he is now in our power. We know that this news will bring you great happiness."

Quoted in Richard I by John Gillingham, Yale, 2002, p222

Philip could now argue that Richard was no longer on crusade (even though he had not returned to his own realm), and that the church's protection of his lands no longer applied.

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Philip in Normandy

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Already, in January 1192, Philip had presented documents to William FitzRalph, Richard’s Seneschal in Normandy, claiming the the lands that Richard had received as Alice's dowry were to be returned to Philip. FitzRalph and the other Norman barons had recognised the documents to be forgeries, and rejected Philip's demands.

After this initial failure, Philip spent the rest of 1192 forging alliances in preparation for the coming war. Crucially, these alliances included one with Richard's brother John. In early 1193, John returned to England with the claim that Richard was dead, and that the crown should now pass to him. Unfortunately for John, the government in England had already learned that Richard was alive and a prisoner in Germany.

Through the campaigning season of 1193, Philip was able to make great inroads into Normandy, including the capture (possibly by treachery) of the fortress at Gisors, mentioned earlier. Importantly, however, Philip failed to capture the capital of Normandy at Rouen.

While this was going on, Richard's government was largely distracted by the efforts involved in raising the ransom to secure his release. They came to terms with Philip in July 1193 Barlow, 1955, [p363].

Philip and John spent the rest of the year in a series of attempts to bribe Henry VI to extend Richard's captivity (or even to hand him over to them!). They failed, and Richard was released in February 1194. Philip wrote to John Saying:

"Beware, the Devil is loose."

Europe in the Middle Ages, Ierne L. Plunket, Oxford, 1927, p168

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Richard Returns to Normandy

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On his arrival in England in March 1194, Richard set about raising funds for the coming war with Philip. By May he was in Normandy.

He immediately relieved the siege at Verneuil, and re-captured a series of fortresses in the south of the Duchy. Richard attempted to engage Philip in battle at Fréteval. Philip and his army fled (in fact, Philip was almost captured), but Richard succeeded in capturing Philip's baggage train which included the Royal archives - including a list of the traitors in the Angevin camp that had agreed to aid Philip against Richard!

A month later, Philip's army re-grouped and routed a force under Count John (who, sensing that the tide was turning, had returned to Richard's side).

The conflict would drag on for another five years, until Richard's death in 1199.

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The Conquest of Normandy

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The conquest of Normany that you referred to in the question actually happened under Richard's successor, King John. John has a good claim to the title of England's worst ever king (and there's some stiff competition!).

Philip captured much of Normandy in the campaigns of 1202-1204. As with so much else in his life, John was not up to the task of defending Normandy. He was even less successful in his later attempts to retake the Duchy. The loss of Normandy effectively broke the Angevin Empire which had been Philip's goal from the start.

John's failure earned him the nickname "John Lackland".

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Sources:

• Barlow, Frank: The Feudal Kingdom of England, 1042-1216, Longmans, 1955
• Gillingham, John: Richard I, Yale, 2002
• Plunket, Ierne L: Europe in the Middle Ages, Oxford, 1927