Whoo. You gave me a run for my money there.

I'll try and cover everything but if I miss something my apologies. I'll come back to it.
In fairness, they haven't had a decent showing in recent years. The the two leagues basically played .500 ball from '03 to '05. In fact, by my math its 378-377 in favor of the NL. Now, if you mean recent by the last three years, that's fair. But from '04 to '05 its 262-242 AL. Which is pretty fair. Its '06 that dramaticaly skews things. And I really don't mean to say that were it not for some bad luck and such the NL would have played .500 ball or won this interleague season. They were a worse division so logically they should have lost. I dare say badly. But 60 games badly? That's a bit much, in my mind.
Interleague just seems to go against the arguement, unless you're speaking in the most short term way possible. But as soon as you call back a decade or so through WS and All Star games it seems fair to do so with interleague (or as far back as it goes). And the reality is that the NL has held the edge in interleague right up until the recent past, even holding the extremely slight lead coming into this season.
Now WS and All Star Games are clearly different. The WS dominance is completely fair. But I do think its skewed a bit by the presence of a Yankee Dynasty. 5 of the 10 AL wins came from one team. That speaks more to the dominance (and dynasty) of the Yankees than the AL, I'd think. The AL still has the edge and all and I wouldn't mean to take away the Yankee wins (they are, after all, part of the AL) but it does seem to push the odds. In the end there's also a lot to be said about that being one series between 2 teams. If the AL team loses to the NL team this year realistically noone will say that proves the dominance of the NL. 8 good AL teams vs 2 good NL teams doesn't mean much when it comes down to 1 vs 1. And since there's no guarantee that the best of either league will be that 1 it pretty much kills a "our best vs your best" idea. We of course like to believe that the best team wins but in reality we all know matchups, seeding, timing, and luck all factor in.
All Star Games I don't even know how to factor in. A collection of stars from each league seems like a very rough way to judge each other. Especially when the selection process is as wild as it is with the MLB All Star Game. Any number of the best players in the league can be left out for fan favorites, player favorites, coaches' favorites, veterans in bad years, etc. And then they come under the control of a manager who is unfamiliar with them, who may or may not have a major intention to win, and who manages in a manner to protect the players from any true harm. If you took the best starter in the NL and the best starter in the AL and set them against the best lineups in both leagues I imagine that would yield different results than letting them throw 2 innings and then turning the ball over (presuming you even get the best starters starting or pitching). If I had more faith in the ALS's ability to judge the leagues I'd even argue that having Mariano in the 9th for the last 10 years has helped the AL quite a bit (although I don't actually know how often he's been used in that capacity).
But you're right. It all factors into perception. The AL dominanting the WS and All Star Games in the recent past help push this idea, if not start it. The DH issue pushes it. And while interleague realistically doesn't show it, surely skewed perceptions do (like looking at just the last year or two or, as I suggested earlier, the fact that the Yankees posses the best record in interleague play).
And certainly, I didn't mean to argue that the perception is pushed by this one year of interleague. All that this season's interleague record did was manage to hammer home to the point that was being said for quite a few years now. And it did it decisively. The problem comes that folks like JJazz (and my apologies, I don't mean to call you out) start to think of that as a further dominance. And yes, the ASG and WS examples exist but it then gets extended and people start to presume that the AL has dominated interleague play as well. It all gets pushed. The NL's horrible 2006 has essentially sabotaged the idea of the NL in the recent past (and probably near future). And yes, the 0-8 record in the WS the last 2 seasons doesn't help either. But I'd argue that is what it is. Two AL teams that were playing absurdly well. Boston won 8 straight and had swept their 1st round opponents. Chicago also won 8 straight and became only the second team to go 11-1 in the playoffs since the division series were added. That was dominance by the individual teams, not the league. That they came in back to back seasons, both for the AL is impressive but surely a bit of coincidence. But yes, it certainly aids the perception quite a bit.
I never meant to argue that perception was unfair. There are a lot of reasons for it and it holds up this season. But it does get exaggeration, and the hard to judge nature of the AL vs NL matchups
To briefly touch on the DH issue, I think that is a 100% fair point. Clearly the NL lineups are weaker than the AL lineups. And it gives AL pitchers an extra batter (who is often a key bat) and more well rounded lineups. NL pitchers get occasional situations to get out of spots and generally get weaker parts of the lineup that can help them last longer and yield better results. Its much easier to kill a rally in that situation, clealry.
That being said, I DO think its gets exaggerated from time to time. For one, NL does play a different style of ball and its not so much "every 9th batter" as it is "every 9th batter for a couple of go arounds and then things change." In some ways I'd argue that NL managers occasionally find themselves in situations that might even favor the situational selection of hitters. There are certainly occasions where you rather have the first bat off a NL bench hitting rather than the 9th batter in an AL lineup. But I don't mean that to be an arguement against the idea. Like the '06 interleague thing I was arguing I just think its dulls, if just slightly, the edge the AL has.
And in the end, really NL pitchers DO get dismissed out of hand quite a bit. And while its fair to say that comparing an NL pitcher's numbers vs an AL pitcher's numbers is hard to do considering the league discrepencies I think there are far too many instances when AL fans immediatelly side with the AL pitcher for no reason other than his league. Or when AL fans dismiss NL pitchers out of hand because they are NL pitchers. There's a difference between saying that NL numbers don't quite match up with AL numbers and saying NL numbers don't matter much at all. And I think the second idea exists in a lot of fans and quite often on this board.
I'm sure I missed something or misinterpreted something. But its late. I just wanted to respond as best I could because I don't think I'll get back to this until late tomorrow. But my apologies for any omissions or mistakes.