Wellington's reputation has of course been enhanced. He never said "I don't know what effect these men will have upon the enemy, but, by God, they frighten me". His real line was, "As Lord Chesterfield said of the generals of his day, 'I only hope that when the enemy reads the list of their names, he trembles as I do.'" Likewise, the full Waterloo quote is "It has been a damned serious business... Blücher and I have lost 30,000 men. It has been a damned nice [i.e. delicately balanced] thing — the nearest run thing you ever saw in your life." In both cases, his admirers (who were and remain many) have made his actual words a bit crunchier for the consumption of posterity. (It should also be noted that the original version in both cases is a bit more modest - in the former, he is actually quoting someone else; in the latter, he concentrates on the massive number of casualties.)
But the line about the stable is rather different - it doesn't reflect terribly well on the Duke, unlike the other two. And interestingly its origin is not with the Duke himself at all, but with another famous orator of the nineteenth century: Daniel O'Connell. Here we have a court hearing in which O'Connell is reported as saying (on 1 October 1843, at a dinner speech after the Monster Meeting earlier that day), "The poor old Duke! What shall I say of him? To be sure he was born in Ireland, but being born in a stable does not make a man a horse."
Wellington was of course born in Ireland (probably in Dublin) and brought up at Dangan Castle near Trim, Co. Meath (see local debate). By 1843 he was 71 and getting on a bit, but nonetheless was still Commander-in-Chief of the British army and minister without portfolio in the cabinet. O'Connell's ally, William Smith O'Brien, speaking on the State of Ireland at the House of Commons on 4 July 1843, had also mocked the Duke's pretensions to Irishness as part of the general failure of British state institutions to represent Ireland: "Again, let us see how facts actually stand: there are Cabinet Ministers, Englishmen 10, Scotchmen 3, Irish O. The Duke of Wellington is so much denationalized, that I believe he scarcely considers himself an Irishman, and certainly cannot be called a representative of Irish interests in the Cabinet." Wellington responded in a somewhat verbose speech a few days later in the House of Lords, attacking O'Connell directly for wasting time on political agitation rather than the real interests of the Irish people.
So in fact, the origin of the quote was not the Duke dismissing his own Irish origins (and in a rather cursory search, I haven't found any strong statement from him on his own Irishness one way or the other) but in fact the radical Irish politicians denying him any right to speak as an Irishman just because he was born there.
The quote has had a long afterlife. Here, for instance, Jim Tully (incorrectly) corrects his former party colleague Patrick Norton (who had just resigned from the Labour Party and was about to join Fianna Fáil) as to its origin in an 1968 Dáil debate on abolishing proportional representation.
Mr. Norton: Deputy Treacy mentioned nationality when speaking of the system of election. I cannot see it is in any way relevant whether the system of election is a British or an Irish system. Like the Minister, I think that, in fact, both the PR system and the straight vote system are British systems. I think it was Daniel O'Connell who said that the fact you were born in a stable does not make you a horse.
Mr. James Tully: Just to get the facts right, it was the Duke of Wellington.
 Mr. Norton: I am not aware who said it, but I certainly know the wisdom of it. I do not think it is important whether the person who conceived the system was English or Irish or what his nationality was. Surely it is the merits of the system we should be concerned about?
Most gloriously, there is an allusion to the quote in Finnegan's Wake. At the end of Part 1, Episode 1, there is a long meditation on "Willingdone" (who combines aspects of both Finnegan and the Duke as commemorated in a (fictional) museum/museyroom under the Phoenix Park monument) which concludes as follows:
This is the Willingdone, bornstable ghentleman, tinders his maxbotch to the cursigan Shimar Shin. Basucker youstead! This is the dooforhim seeboy blow the whole of the half of the hat of lipoleums off of the top of the tail on the back of his big wide harse. Tip (Bullseye! Game!) How Copenhagen ended. This way the museyroom. Mind your boots goan out.