Muslims hold the view that every nabi (prophet) is a rasul (messenger); and that every rasul is a nabi. As a matter of fact, those two words signify two aspects of the mission Allah Almighty has entrusted to every prophet and messenger; and we see no example or evidence in the Quran where one aspect alone is present in a person chosen by Allah as His Messenger or Prophet.

There are Muslims who argue that 'rasul' means one who was given a Book of Allah, or brought a Shariah or code of laws. However, I find that this doesn't explain why Prophet Ismail (Ishmael, peace be upon him) has been called a rasul in the Quran.

See this verse which reads what means;

*{And relate the story of Ishmael as mentioned in the Book. He was indeed true to his promises. And he was a Messenger, a Prophet.}* (Maryam 19:54)

The differentiation between the office of "prophethood" and "messengership" in a prophet is not really warranted by the Quran. Because, from the Quranic point of view, every rasul is nabi, and every nabi is a rasul. These two words are often used as interchangeable terms and represent two aspects of the same office and two functional responsibilities entrusted to the same person.

A Divinely appointed reformer is a rasul so far as he receives revealed messages from Allah; and he is a nabi in the sense that he conveys those messages to the people to whom he is sent.

Thus every rasul (messenger) is simultaneously a nabi (prophet), or has to perform as one, because after receiving Divine messages he conveys them to his people. And every nabi has got to be a rasul because he needs to convey to his people those messages which he has received from Allah Almighty. Only the functions of nabi follow those of rasul. First in his capacity as rasul he receives messages from God and then only in his capacity as nabi he conveys them to his people.

This is why everywhere in the Quran when these two words rasul and nabi occur in close conjunction, invariably the word nabi follows the word rasul because that is the natural order. For instance, see the verse:

*{And relate the story of Moses as mentioned in the Book. He was, indeed, a chosen one; and he was a Messenger, a Prophet.}* (Maryam 19:51)

That is to say, Moses was chosen as messenger first; so he received revealed messages from Allah Almighty; and then he conveyed those messages to his people and thus performed as a nabi.

In the foregoing paragraphs we have seen the difference in meaning between the words, 'rasul' and 'nabi'. Also we saw how those two words could be used interchangeably to refer to all the prophets of Allah. So, the answer to the first part of your second question is clear. We may say the first rasul and nabi was Adam (peace be upon him).

About the hadith quoted from Abu Dharr Al-Ghifari (may Allah be pleased with him), one version of the hadith reported by Abu Dharr is this one, though there are other slightly varying narrations:
 
"The Prophet (peace be upon him) was asked about the number of the prophets and messengers. He replied that there were 124,000 prophets among whom 315 were messengers." (Ahmad)

But scholars are of the view that this hadith is weak. And if we go by the words of this hadith, they would imply that there were prophets who did not receive messages from Allah, which is not acceptable. So, the distinction made between prophets and messengers in this manner has little basis.

Either way, Allah the Almighty has commanded us to respect and honor all His prophets alike, without making any distinction between one and another of them:

*{Say you: "We believe in Allah, and the revelation given to us, and to Abraham, Ishmael, Isaac, Jacob, and the Tribes, and that given to Moses and Jesus, and that given to (all) prophets from their Lord: We make no difference between one and another of them: And we bow to Allah (in Islam).}* (Al-Baqarah 2:136)

Also see this verse:

*{The Messenger believes in what has been revealed to him from his Lord, as do the men of faith. Each one (of them) believes in Allah, His angels, His books, and His messengers. "We make no distinction (they say) between one and another of His messengers." And they say: "We hear, and we obey: (We seek) Your forgiveness, our Lord, and to You is the end of all journeys.}*  (Al-Baqarah  2:285)

The foregoing verses mean that we have no right to differentiate or categorize Allah's prophets and messengers as to their degree, status or performance; nor should we discriminate one against another. Instead, our duty is to respect and honor them alike.