Depends on the language I guess. When I hear songs in thorny languages such as German or Polish they can touch and move me only if the music is outstandingly beautiful and completely surpasses the text. On the other hand when I hear Brazilian songs I feel like the words caress me, the language itself is so soft and musical that the meaning of the words doesn't matter, I'm immersed in a bath of honey and can imagine they say whatever I want them to say.

Thanks for the link. I think one of the strangest anachronisms in the video you posted is how little the singer cares about the presence of the camera. She's totally focused on her audience and her performance.  

Responding to the song -- it's curious but even though one is aware of not understanding the words she sings, one is immediately reassured by the accompanying instruments, which nudge and echo and, universally, speak a very familiar kind of language. I could listen to her all morning. Still, given the care she takes with her words, it's hard not to feel one is missing something; she is obviously very articulate. 

Is this so different from the way I feel hearing songs in my own language, just out of articulate range? 

Thinking about my own emotional response, for your original questions, I would distinguish between two very different cases: cases where it seems that the singer has something very precise to say and is saying it in words you do not have access to, whether because of vocabulary or ambient noise, and cases where it seems the singer is singing because they are too overwhelmed to speak and are grasping at whatever words happen to be near (this second is essentially Lorca's description of canto jondo).  

For instance the first minute or so of this video, starting around 30 seconds:

http://www.youtube.com/watch?v=nx251wVaj8E

Foreign words in songs: a very charming subject. I used to listen to Bach's St. Matthaus Passion, without understanding a single word. Then I moved to Germany and one day I had this amazing experience, this epiphany: suddenly I could understand what the words meant! I adore this piece of music, always have. And words used to be just part of the instrumentation to me. Well, the new experience was, after the first surprise, a bit disappointing: I felt like I was loosing part of the musical complexity by gaining useless information (I really don't care about the literal message). Very strange.
Again, I know some hebrew but not enough to understand what was recited in prayers in my jewish upbringing. This contributed to my acceptance of this upbringing, to understanding it more as a bunch of family traditions rather than a religious education. Could I understand what I'm saying when I sing at Passover, I would probably feel very estranged and uncomfortable.
Thirdly, would I find cheap pop music so annoying, if I couldn't understand the lyrics? All this "love" rhyming with "dove" makes it very difficult to me to concentrate on the musical aspects.

On a positive note, I loved your video, Roy, and I enjoy when enunciation becomes a means of conveying musical  structure. I'm not sure if I would enjoy it more if my hebrew were better...