I've got a few questions that need to be answered.

I just recently beat the Dragon God. Wasn't that hard actually. So after I killed him I started exploring and I came across stones blocking a path. And I saw a sword. Is it true that I need to have pure white / black tendency ? Or do I need some certain weapon ? 

To get that sword, called Dragon Bone Smasher (My personal favourite), you'll need pure White World tendency on world 2. 

After I beat the Flamelurker I headed straight to Blacksmith Ed. I talked to him a couple of times until finally he asked if he could get the Flamelurker soul. I agreed to give it to him. Do I need to give him other souls in order to make armor, weapons, spells, etc ? Or will he give me whatever he made with the Flamelurker soul after a period of time ?  And what about the souls of the Adjucator, Tower Knight, etc ?

The Flamelurker's soul enables Blacksmith Ed to use bosses souls to craft special weapons, such as the Lava Bow with the Armoured Spider soul or the Scraping Spear with the Phalanx Soul. To get those special weapons, you'll need to level up certain weapons to a certain amount of (Plus) levels. For example, to get a Lava Bow, you'll need both the Armoured Spider soul and any kind of Bow leveled up to Plus 7.

I've seen my friend kill Ostrava of Boletaria. And he got the mausoleum key which allowed him to face Old King Doran. I can't find Ostrava roaming through 1 - 1 or 1 - 2 but I do see him sitting down in the Nexus. Does that mean I have to kill him in the Nexus ? Or does he roam around in 1 - 3 or 1 - 4 ?

When you rescue Ostrava in each individual stage, he won't appear in it again. Instead he only appears in the next stage you haven't completed. The quest ends at 1-4 and you'll get the Mausoleum Key there, but if you're unfortunate and he dies somewhere on 1-1, 1-2 or 1-3, you can still get the key if you can locate the place where he died (of course, if he died off a cliff or something, you won't get the key in the current playthrough).

I just realized that I don't have the complete set of Brushwood armor. If I invade someone else's world as a black phantom, will I still be able to get the armor if he didn't get it ? Or do I need to do a new game ?

You can get a full set of Brushwood armour in two places on world 1. In world 1-1, both in pure White or pure Black World Tendency, the door leading to the side area near the start will be open. In there, after ascending a couple of stairs, you'll fight off against either Executioner Miranda (Pure White) or her Black Phantom form (Pure Black). In the place where she was hiding, at the bottom of a long ladder, you'll find a body of a soldier, who carries a full set of Brushwood Armour. Make sure you have enough weight available to carry it.

The other full set can be obtained from Biorr, after you rescue him from the prison in 1-2. If you let him die either in the fight with the Penetrator or the Blue dragon, you can loot his body for the set. 

If you don't get any of those sets, then you can only obtain it again in a new game plus.

How do I get all the trophies that relate to obtaining the best weapon by ___________ ?

You'll need to level up the required prefix weapons to the maximum level. Hardstone and Sharpstone upgrades go to plus 10 and every other upgrade goes to plus 5

If you can answer any of these questions, it would be appreciated.

Damn. Thanks for all the help. And I would be very grateful if you could tell me what item was on the dead body on 1 - 2. There's a door located right before the Tower Knight fight. I opened it and I saw one of those clowns. After killing him I opened the other doors and went down the tunnel. Killed one of those silver lizards and their was a body with a item. But when I picked it up they said I was overcumbered. Was that a rare item or just like a shard or something ?