While it’s a good game, I think Borderlands completely fails the OPs tests:
#1: Okay, fine it does pass this one. 
#2: Fail. FPS games with twinstick controls are the antithesis of “stuff you want to sit your non-gamer friend down with”
#3: LOTS OF SHOOTY DEATH.
#4: HAH
#5: Nope.
OP: Do not buy Borderlands to play with your girlfriend.
Let me add Sacred 2, then. I played this a lot with my wife on my PS3, and just saw that its also available on the 360. Its a loooong game, but would fit his requirements nicely.
My wife isn’t a gamer and she played threw fable 3 twice, we’ll past when I gave up playing with her.
Castle Crashers satisfies all your criteria except having a great storyline, and it’s a lot of fun.
A lot of people didn’t like this game for some strange reason. My brother and I love co- op rpgs. Thisoneis rated M but it’s mostly just for crit hits. There is decapitation and some blood but as far as playing a rpg together it’s one of the best. Very fun leveling looting upgrading gear. A fantastic story. I definitely recommend this game to any rpg fan who likes to play co-op
Which game are you actually talking about, since there are a bunch of LOTR spinoff games?
Second this. They’re definitely $20-level games, as you can blow through the campaign in one or two evenings. The rest of the game is grinding unlocks. Still, the campaigns are pretty fun, and the gameplay is really easy and forgiving.
I played through this with my wife, and she liked it (although this was on the PS3, I know it’s available on the 360).
We tried to play Sacred 2, but the co-op was massively buggy. Again, though, this was the PS3.
I’d would recommend Skylanders. Whilst it’s advertised as a kid’s game with a lot of extra characters to buy, you don’t have to buy anything other than the game itself as it comes with the portal of power thing and 3 of the figures (including Trigger Happy, who’s the best one by a distance anyway) and it’s actually a fun game that stands up pretty well. It’s not difficult, but you said you didn’t want difficult.
And, as other’s have said, the Lego games are great.
I know it doesn’t fit the OPs bill, but my g/f loves it. We started playing Left4Dead and didn’t like it much…honestly not sure what makes it such a great game.
We’ve been playing Borderlands like fools for the last couple weeks. You shoulda seen the look on her face when she learned there’s a Borderlands 2 
Are there any other games like the Lego games? My wife and I, who only have a PC, have enjoyed all the Lego games and actually did play X-men Legends 1 and 2 on the PS2.
I wish we had more co-op games to play; we love 'em.
For PC, Lego Lord of the rings is $7 on Steam until Thursday.