Is it possible to talk to Esbern if I don't kill Paarthurnax in his quest? I went to talk to him and he assigned me the quest and refused to talk to me! Is there a way not to take this quest so I'll be able to keep him and the graybeards friendly?

no. its like the stormcloak vs imperials. u have to choose a side.

note: even if u do nothing, ur choosing the greybeards side by not killing paarthunax.

If you do choose to kill him though, atleast it adds a LITTLE bit more onto the story line of the game

So it's "Kill the drake or get kicked"... I would have liked it more if there was a persuade option!

If you are on pc you can use console to make the blades think that you have killed him.
Here is how to: http://www.gamespot.com/the-elder-scrolls-v-skyrim/forum/messages/platform/pc?topic_id=m-1-61097860&pid=615805

Are you guys saying it's either the Grey Beards OR the Blades?

At the point I'm at it looks like the only Blade is that one chick in Riverwood. That's easy.

Simple. Keep Paarthunax. I got a boost for Fire Breath and the greybeards still help me. The blades are useless