I'm confused. I did the quest and I obtained the Ebony blade given to me by the whispering lady who ID herself as the Daedric Lord Mephala.

Although no vocal agreement was made. By taking the blade and start fighting with it does that count as making an agreement with Mephala? Will my soul go to her when my character finally dies?

Your character signed a pact with Mephala by using the Blade, but your characters Soul is bound to Akatosh.<--- Forum Consensus has decided Akatosh has first dibs, and Sheogorath or Sithis are in a heated race for second.

Why would Sheogorath have any tie to the dragonborn's soul? And what about the Thieves Guild daedric prince?

Pretty much all you need to know is that Akatosh has dibs on your soul from birth because as Dragonborn, you are born with the blood of a dragon. Dragonborns wouldn't exist if not for the pact Akatosh made with St. Alessia, so there you go.
All the Daedric Princes keep souls of their worshippers/faithful/champions if they can help it. Some of them are just more concerned or get more of a right to one's soul over another prince's, like how Nocturnal takes Nightingales' souls or Hircine takes werewolves.

Well, no one said you had to start fighting with it, did they? My character took it mostly to get Mephala to leave poor Nelkir alone, then promptly took the blade and locked it away in a chest in one of her houses.

If you don't actually use it, you're not making any bargain with Mephala. ...You just stole her artifact and probably ticked her off.