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Calculating elo ratings between 2 players is fairly straightforward. See http://pastebin.com/9sjQyxiU.

The question is, how should the elo ratings be calculated for matches with 3 or more players?

3rd place = -1 which is loass
2nd place = 0 which is like a draw. Which still can change a players rating
1st place = 1 which is win

4th place = -1
3rd place = -.3333333
2nd place = .333333
1st place = 1

and so on.....

Hmm.. Wouldn't it be better if 2nd place got 0.5 and 3rd place got -0.5 in a 4-player match? This way, in a 5-player match, the 3rd place would get 0 and 4th place would get -0.5?

Also, what do we do about opponent's rating? The W's above work with a 1-1 calculation. So, if we assume Player 1 won, would we use the average rating of the other players? The max rating?

well there would be a 1 point gap between 2nd and 3rd.....and then only half point gap between 1st and 2nd ...and 3rd and 4th. Could do it like that but makes more mathematical sense doing 1/3 of a point and negative 1/3 of a point.

I don't have a good understanding of the basic elo system to comment on part b of you question

How about this? For each player, sum the Elo points that would be won/lost across all 1-1 pairs with that player.

For example, suppose Players A, B, C, D come in 1st, 2nd, 3rd, 4th. A gains points equal to the points he would gain for beating B, C, and D in individual 1v1s; B would gain/lose points equal to the points he would gain for beating C and D minus the points he would lose for losing to A; etc.

Pros:

Bigger rating moves for bigger games - Suppose all players have the same rating/skill. In 1v1s, a player would on avg gain X points 1/2 the time (1st), whereas in a 4-player game, a player would on avg gain 2X points 1/2 the time (avg of 1st and 2nd).

Reduces to 1-1 Elo in boundary cases - Suppose there are three players, A B C, and C has a rating of -Inf. We can assume that C will always lose. A and B will not gain any points from beating C. Hence, the system reduces to 1-1 Elo between A and B.

This is actually how the rating was calculated - I recently modified it so that the the highest ratings, of the players you beat and lost to, are used. This allows for more 'positive' rating changes :)