Can I get a stealth check for sneaking up? (It'll only determine whether or not he is surprised, the attack will still hit regardless)

Vaelin

Stealth: 24

Init: 22

You manage to get the thug by surprise, and your shocking grasp sends enough electricity through his body to kill a peasant. He arches up in pain, but he isn't dead. roll for initiative.

Thug: 2

Since he was taken by surprise, his turn will be skipped until round 2.

Yutari's Initiative: 13

Can I use my turn to go get my rapier?

Ok, here is the initiative list:

1. Vaelin (He already took his turn initiating combat)
2.  The thug (currently surprised, so we will skip his turn)
3. Yutari (current turn)
4. Finnian

Vaelin is behind the thug, on the right side of the room. The thug's side is exposed as Vaelin attacked from the back, his hand transferring an electric shock to the thug's back. The smell of burnt leather and chicken fills the air.

You can definitely go to pick up your rapier, but picking up a weapon would cost no action.

Yutari will go pick up her rapier then take a stab at the thug.

Attack Roll: 15, Damage: 11

If it does hit, then Yutari will divine smite - 8

Vaelin will attempt to shock the guard again (12 to hit with 6 damage)

As the thug arches back, Yutari comes in from the side and stabs the thug, her rapier going in between the ribs, glowing with divinity. Finnian than comes in, and goes to stab the thug. This damage will reduce the thug to 0 hp, so I'll let you choose how to finish off the thug, whether to nock him out or kill him, and if so, how you kill him.

You did

He checks the guard. 
he also retrieves his own items. 

The guard has the keys which go to the chains which once bound you three, a mace, 4 Silver, a crumpled piece of paper, a heavy crossbow, as well as 3 crossbow bolts. He also wears leather armor.

It has the word "eggs" written on it 

"May I suggest, we get out of here before anyone else comes along." Vaelin suggests.