What happens if you kill the Pursuer during your first encounter?
First of all, that is your only chance to kill him there. If you kill him then that counts as the kill of the boss and he WILL NOT appear at the top of the castle.
What happens if you don't kill him in your first encounter?
He will not appear again in that area and you will have to face him at the top of the castle.
If you were unable to kill him during the first encounter, is it possible to gain the first encounter kill rewards in some other way later on in the game (vice versa)?
As far as I know, he drops the exact same loot as if you faced him as a regular boss.
I mention his drops in the next lines, so I guess it counts as a spoiler.
He drops Soul of the Pursuer and The Ring of Blades whether you kill him as a
regular boss or when he is dropped by the eagle.
First of all, you need to know that dex scaling is pretty wonky. There's not much benefit between an A and S rank. Keep that in mind.
Also note that poison and bleeding scales with DEX.
The weapons with best DEX scaling ( A/S ) are:
Ricard's Rapier (starts with A scaling, gets S once upgraded to +5) - Huntsman's Copse. Bottom of the pit after first bonfire.
Rare drop from Spiders in Brightstone Cove Tseldora.
Curved Dragon Greatsword (S)- Trade Ancient Dragon Soul with Ornifex
Hunter's Blackbow (S) - Found in the chest down a ladder at the second bonfire before the Dual Dragonrider boss at Drangleic Castle.
Mirrah Greatsword (A) - Buy from Merchant Hag Melentia in Cardinal Tower or Majula after killing Lucatiel. Given to you in Aldia's Keep at the first bonfire by Lucatiel after following her storyline.
Warped Sword (A) - Trade Flexile Sentry Soul with Straid of Olaphis
Espada Ropera (A) - talk to Chancellor Wellager in Drangleic Castle after defeating the Giant Lord to receive the Espada Ropera and the Royal Dirk.
Caestus (A) - Purchasable from Merchant Hag Melentia
Parrying Dagger (A) - Located in a chest behind a boarded up wall closest to Cell bonfire in Lost Bastille head left from the bonfire and head up the stairs and down the rampart. Go down the ladder to your left and go through the door way and head right. There will be a boarded up door. Break the boards and head through and to you left. You will see a door. Use the Antiquated Key to open it and head straight and through the doorway, climb up the ladder to your left and drop down onto the raised platform. There is a passage way that leads to 3 chests. It resides in one of them.
Spider Fang (A) - Trade Soul of the Duke's Dear Freja to Weaponsmith Ornifex
Uchigatana (starts with B, gets A on +5) - Purchase from Steady Hand McDuff
Washing Pole (starts with B, gets A on +10) - Dragon Shrine (Chest) (Mimic)
Chaos Blade (starts with B, gets A on +5) - Trade the Old Witch Soul with Ornifex
Blacksteel Katana (stats with A, gets S on +1) - Dropped by Alonne Knights at the Iron Keep
Manslayer (starts with B, gets A on +5) - From the rise of the dead bonfire in the Shrine of Amana go up the stairs enter the door. Make sure you are hollow and you rescued all sirens or the door will not open. From there continue along the path and through the shrine room and down the next set of stairs. Now go outside and take a left and walk along the ledge. The katana will be inside a chest to your left near the wall.
Darkdrift (starts with C, gets A on +10) - Talk to Grave Warden Agdayne in the Undead Crypt after obtaining the King's Ring.
Murakumo (starts with B, gets A on +2) - Bought from Ornifex
Arced Sword (starts with B, gets A on +2) - Trade Flexile Sentry Soul to Straid of Olaphis
Spider's Silk (S) - Trade weapons Ornifex the Soul of the Duke's Dear Freja.
Syan's Halberd (A) - Dropped by the Knights wearing Syan's Armor(gold) and wielding halberds in Drangleic Castle
Best Answer
In Dark Souls 1 you could attack from behind shields with spears and rapiers this has been transferred to the second game...
The big change in Dark Souls 2 for weapons of this Nature is the Halberd you can use your Halberd as you would a spear from behind your shield this however changes the type of attack to thrust as apposed to slash so be aware of the enemies resistance to stabbing attacks. As for other Weapons that can be used from behind the shield there are Rapiers and all other types of spear and halberd with a point.
As for your Stamina you can find many ways of increasing the recovery speed such as The Cloranthy Ring and green blossom.