The skaters perform their LP in the qualifying round. Occasionally, they may perform a different LP (Stephane Lambiel's doing that - he's switching them from Euros, so the one he performed for the LP there is the qualifying skate for Worlds, and vice versa. IIRC Plushenko has done that before too). Most of the time though, they perform the same programme as in the LP section of the competition.
The qualifying groups are decided based on last year's results, and are alternated between two groups. So, for example, in Group A for ladies qualifying will be Kwan (1st last year), Suguri (3rd last year), Volchkova (5th last year), etc. In Group B will be Sokolova (2nd last year), Cohen (4th last year), etc. Sarah Hughes who came 6th last year isn't competing, so the next skater in Group B will be the 7th place skater from last year, then it carries on like that.
Once all the placed skaters have been assigned, I believe the rest of the skaters (including those who didn't compete last year, such as Slutskaya) are assigned by country, again alternating between the two groups. I believe they start going through the countries alphabetically (starting with Australia), but they might choose a country at random to start from, then continue alphabetically through the countries/skaters. So for instance, say there's three skaters from Australia - skater 1 would go in group A, skater 2 in group B, skater 3 in group A, then the next skater from the next country goes in group B, etc.
The purpose of the qualifying groups is to streamline the competition a bit - that is, you have to place in the top section in order to qualify for the short programme. I'm not sure how high you have to place, but there's a certain cut-off point in the rankings that you have to reach/be higher than in order to stay in the competition for the next part (the short).
After that, you have to place in the top 24 in the short programme in order to advance to the long programme.
The qualifying round didn't used to count towards the final score, but I believe they changed it in something like 1999 so that the results would count.
The results of the qualifying go towards how the short programme groups are put together. I *think* the final short programme group is made up of the top 3 from each qualifying group. 4-6 from each Q group are then put in the penultimate group for the short, etc. The overall results after the short are then used to assign skaters to each group for the long programme (the top 6 after the short - based on results from both the short and the qualifying - go in the final group for the LP).
All the results are used together to work out the final results (e.g. a skater who comes last in their qualifying but wins the short and the long probably won't win overall, because of their qualifying results).
Because the pairs competition doesn't have as many entries as the singles categories, that's why pairs doesn't have a qualifying round - it's not necessary.
Hope that clears it up a bit!
