Kamiel, it is OK to play the Urussov. But how do you avoid the Italian after 1.e4 e5 2.Bc4 Nc6 ?

That is a most remarkable website to which you refer .  However, I would not trust 14. Qb3 "!" until I had analyzed it very thoroughly.

Assuming that it is not unsound  , I think that it would fine to play the Moeller Attack, which is great fun.  Implicitly, of course, you offer an easy draw to anyone who is well booked, but perhaps in a club setting, or even in a Swiss against a given opponent, that is not so bad.  Bear in mind that in a Swiss event, an opponent who expects to beat you is most unlikely to play into the Moeller.

From a practical viewpoint, I would think that it would be good for an Italian player to have BOTH the Moeller and the 0-0 idea in his arsenal.  Then depending on how he felt about the game when he reached the crossroads, he could choose either.

1) There is one forced draw 13.0-0; not numerous. And it is almost never seen any more in recent years since Portisch reinvented 13.h6 in 1968 .

2) Theory ends at move 14 of your line not cause it isn't chewed out but cause there's nothing left to analyse after that. White has verry little attacking prospects and I really doubt any strong player would prefer to play with the white pieces in this position, all sources I have read  say this line is just the refutation of the 6.0-0 ... 9.Qe2 line as it gives black a comfortable and better game. I have even read Emms analysis about this line and as far as I remember it was just 1 small page long just giving games with commentary of Burgess and almost no analysis of Emms himself, and even this repertoire book suggested that black was just fine. About you playing correspondence chess, I'd really like to see some of your correspondence games in this line (ending 13. Bf5) where white is victorious as I really doubt that black can get a worse position if he is allowed computer assistance.

Well ok, first it is true that it is believed to be analysed to death for the biggest part but now Qb3 seems to be an improvement in the most important line of the moeller attack so there might still be some spirit in it. And even if it is chewed out, isn't essentially the same true for the kings gambit, vienna gambits and numerous other openings?

But for your line (which is probably even more chewed out as the moeller attack in my opinion as the most interesting main line 9.Qe2 hasn't changed for over 100 years), except if you can give some kind of improvement in the following line, then I don't really see why white should play it.

1. e4 e5 2. Nf3 Nc6 3. Bc4 Bc5 4.c3 Nf6 5.d4 exd4 6.0-0!? Nxe4! 7.cxd4 d5! 8.dxc5 dxc4 9.Qe2!? (after Qxd8+ Kxd8! the best white can hope for is that he can draw the ending)  Qd3 10.Re1 f5 11.Nbd2 0-0 12.Nxe4 fxe4 13. Qxe4 Bf5. 

This main line has already been played more than 100! years ago and leads to a final position that is slightly better for black as practice has shown.

So either you have some nice improvement, then please share it or else explain why it is so interesting as the best white gets is a bad position.

1) Well if you don't want to take the small risk in the 13. ...  0-0 line, you still can choose go for the drawing line with 17.Qh7+ and even then black can go wrong althoug I've noticed that above 2000 everybody knows how to draw it.

2) 6.e5 is known to be bad for ages, this is what steinitz wrote about it: "On general principles this early advance of the e-pawn ought to prove disadvantageous since the reply [6…d5!] soon allows Black the important majority of pawns on the queenside and also first option of breaking through by …f6 subsequently”.

I'm no expert on the Ne5 line as it has very rarely been played against me and those who did didn't know theory and lost to the 11. ... Ncd6 trap . 

But I'll just copy paste here what Tim Harding writes about in his "the kibitzer" column about this opening.

http://www.chesscafe.com/text/kibitz70.pdf

First he gives anand game but after it writes:

However, White can do better; this game has no real theoretical significance. In Klovans-Estrin, White played instead 14 Bg5 f6 15 Bf4 Nf7 16 d6 cxd6 18 Bxd6 Re8 19 Qd5 Ba6 20 Rfe1 with about equal chance. Estrin's comment was that the white activity and Black's
extra pawn balance one another. Your computer will probably claim an edge to Black. Estrin's book also mentions the possibility 13 Qf4 (instead of 13 Qd3) 13...Ne8 14 d6 (offers a second pawn to interfere with Black's development) 14...Nxd6 15 Ba3 b6 16 Rfe1 Bb7 17 Nd4 with compensation (N.Kopylov-Levenfish, 17 USSR Ch 1949).

To see very extensive analysis about the Ne5 line go to 
http://sorrowdy.mysteria.cz/15.htm a fantastic website about this opening.

You ask: what is so attractive in the moeller attack for a gambit player?
I'll answer:
1) If black doesn't know 13. moves of theory then it is a sure win for white often in beautiful fashion.

2)If black does know 13 moves of theory and plays 13. ... h6 then he will be met by the theoretical novelty 14.Qb3 when he will most likely think it is the same as 14.Qe2 line and play 14. ... hxg5 15.Re1 Be6 and get a bad position after 16. Re3.
The other move he might play is 14. ... 0-0 after which  to position is dynamically equal and white gets compensation for the pawn (isn't that what a gambit player wants?) where any attacking minded player would surely prefer to play with the white pieces.

3)If black does know 13 moves of theory and plays 13. ... 0-0 which black things leads to draw (so a strong player won't play this anyway) then white plays 14.Nxh7 and then black can either play 14. ... Bf5 (which is not seen very often, now white has the option Rxe7 which is draw but white can also play Rh4 which leads to wild attacking positions of which you would think a gambit would be happy to get)
the more often seen option is 14. ... Kxh7 15.Qh5+ Kg8 16.Rh4 f5 and now black in general knows how to defend himself against 17.Qh7+ which is forced draw BUT white can still try to win with 17. Rh3 or 17.Be2 which probably with correct play lead to draw too or play the refuted but long forgotten 17.Re1 (this move was long thought to be the refutation of 14. 0-0 so black won't possibly find the refutation against it over the board) . Now unless black knows 60 year old books on the italian game right out of his head (which is very, very unlikely) it is absolutely impossible that he will know how to defend himself. So 1)black does almost  never play 13.  0-0 (90% plays h6). 2)If he does he will not draw but he will lose, I have only met it in blitz games and won nearly all of them except a few were I played Qh7+ after which they knew how to draw most of the times. The only time in a serious game that I saw somebody get so far as 16. ... f5 the white pieces were played by a friend of mine (2100 elo) against another 2100 with black; my friend played 17.Re1!? and black started to think for over an hour played 17. ... Bd7? and resigned after 18.Re6! 18.Be8 Qh7+ 19.Kf7 Rhh6.

So to conclude in what is so attractive in this line; of all the games I played on the internet (mostly blitz) where I reach the position after 7. ... Nxe4 I won no less than 85% of them. This is what makes it attractive to me.

About the 6. 0-0 line you suggest, I agree it is an interesting gambit cause it is not very well know on club level. But I don't really know what you mean with the "underestimated" part of your sentence, as also this gambit has been analysed more or less to death and leads to a good game for black. The best white seems to be able to get is a worse endgame in the 9.Qxd8+ lines or a middle game where black seems to have the advantage cause better placed pieces in the 9.Qe2 Qd3 lines. So I don't really see what would make this gambit so attractive for the gambit player, it doesn't even lead to very wild positions.