Okay...for example...I have been reading some Dark Knight fan fiction (which is different than Batman fan fiction) and there is a whole subgenre of DK fanfic that is about Rachel Dawes hooking up with The Joker. Perhaps this example is unique to DK fanfic, but why is that? It seems like such a strange pairing to me, and yet people write about it all the time.
Its a personal choice if people want to write pairings like this, so its hard to come up with a 'reason why'. People write FF for a number of reasons, usually out of enjoyment for the show/movie/series. As Marlene Arpe says "
In the world of fan fiction, great TV and movie characters never die. They just get new scripts"
FF also, like Celia has said allows people to play around and develop their writing without having to invent all the characters etc. Some people make up OC (original characters) which are completely new but usually aren't done properly or become mary sueish. That being said, when i was younger (young teens) i loved reading xmen ff, and a writer developed an amazing villian, who was an amazingly complex character (and to this day i still remember the characters name).
In regards to pairings, there are canon (Rachel and Batman/Rachel and Harvey) and non-canon (Rachel and ?). Some sites have rules/regs about their FF and only want canon pairings while others don't have any restrictions. I guess people play around with the idea of different relationships and decide if they can apply to the universe. Generally i'd say most non-canon relationships are due to 'reading between the lines' or deciding to push the boundaries for fun.
The important thing is that FF has a 'don't ask won't tell' policy with authors. There have been a number of instances when fans have contacted authors with fan gifts, just to have their fanfic sites etc. shut down for copyright. Not because the author was anti-ff but because of legalities about copyright.
and of course there are instances where fans will step over this line, and actually try and publish themselves off the back of anothers work eg. Russet Noon.
Heres something i posted before about pros and cons of
FF maybe it will help with some of your questions? I have a load of references at the end with links.