Oddly enough, one of the very few things I thought(?) I knew about NFL is that the range of skill difference between the top teams is actually way too high. As evidence for this the article stated how the top two seeded teams nearly always end up in the final, or at least one of them does or something like that. To have a range of elo greater than 400 between the best and worst teams in the same division would mean a score of less than 10% for the lower skilled team, which would be really excessive.  

http://eloratings.net/ is okay, however I've a real dislike of how they've taken to putting a team's "rating" taken from that site on the wikipedia page for the national teams. There's nothing official or even "good" about that rating website, the single claim to fame it has is that it's been mentioned in a bbc article or something the odd time. It's a glorified calculator. 

Also 1v1 elo won't work very well for anything where you're in a group system. For example it doesn't know it's three points for a win, it doesn't know how much weight to put on goals for, it doesn't know whether a team is already through and fielding their B-team. Does it matter whether a team won every match or if they came first in their group and drew one? It's the same final result. They could implement these considerations into the system if they wished. On the other hand they make lots of other modifications to the system on a 1v1 basis, which says to me they don't really understand what they're doing. For example they take amount of goals scored into account outside of how it changes their position in the group. This is in my view like saying someone mated in under 20 moves, or someone who captured their opponent's queen. Sure it is I suppose more likely the person who mated in 20 moves did better than who did it in 60, but it's just not right since a win is the same either way. Some people might be remarkably good at mating quickly, some teams might be very good at scoring goals against weak opposition. I think it's sacrilegious to the whole idea of elo. When I say "they" I of course mean one guy with a website.  
  
They also take friendlies into consideration, but fudge it so it doesn't matter "as much" by reducing the k-factor, which seems very wrong to me. They also count a win as being twice as valuable as a loss.... even though a team scores three times as many points in a group system. Many of these things could be worked on or fixed to make much more sense, however the author of the website seems unwilling to enter into any dialogue or make any effort to explain it. I think the ratings given on the website are misleading and shouldn't be given whole pages on wikipedia or put next to teams. A good elo website would be where you could choose the factors that you believe are important and calculate the ratings if you change it.   
  
FIFA used to have a ridiculous rating system, however I believe they've improved it significantly and it is now much more elo-like in how it works.   
   

Strictly speaking it's not the k factor that determines the spread; that just determines how volatile the ratings are. (Of course a gigantic k will make ratings jump around like crazy and increase the spread that way, but that's just breaking the system.)

538 does use the same rating scale for their NFL ratings as chess does (it's the 400 in the formula on this page explaining their system). They do carry ratings over from year to year, which offsets the small number of games per year somewhat.

But really the main thing causing the small spread is just that everyone in the NFL is really, really good at football. If you restricted Elo ratings to people who are really, really good at chess, say IM level and above, you'd get a spread of only 500 points too.

Yes, Elo is used in many contexts besides chess.

Note that the NFL is a professional league consisting of the best football players in the world. In that context, it should be a bit less surprising that the spread is "only" about 500 points. If the NFL allowed young children to play, the spread of ratings would likely be significantly wider.