from the demon's souls wiki. here is a description of the branching upgrade options for weapons.

• Sharp - Damage bonus is heavily modified by dexterity.
• Quality - Moderate damage bonus from strength and dexterity.
• Mercury - Poison special effect.
• Fatal - Critical special effect.
• Crescent - Damage bonus heavily modified by magic. Minor MP regen.

crescent is HIGHLY powerful if your magic stat is high. My Crescent Falcion +4 does 101 base physical + 109 base magic + 75 bonus magic damage (because of my magic level) per hit. The only bad part about some branching upgrades is that they take rare metals for upgrades that are not easy to find. a lot of these special upgrades must be done by the blacksmith in 2-1.

i killed that bastard right after i released him from the cage.  it's the bug....

regarding pure white offline (from http://demonssouls.wikidot.com/world-tendency)

Getting Pure White World Tendency in offline mode

*Do not enter online mode if you want to follow this guide. Stay completely offline!

• Keep at least 1 Boss alive. Usually the 2nd Boss in World 2, 3, 4 and 5. The 3rd Boss in World 1.
• Clear the path to the Primeval Demon and unlock all shortcuts necessary.
  • World 1 - Kill the Red Dragon in 2-1. Kill Tower Knight in 1-2. Unlock large gate in 1-3.
  • World 2 - Kill Armored Spider in 2-1. Activate Fog to tunnels leading to Flamelurker.
  • World 3 - Kill Fool's Idol in 3-1. Gather all required Keys to reach Lord Rydell's location.
  • World 4 - Kill Adjudictator in 4-1.
  • World 5 - Kill Leechmonger in 5-1. Lower the drawbridge on the cliff. Activated the First and Second Fog in 5-2.
• Now, prepare yourself because you are not going back to the Nexus for a while.
• Get 5-7 Stone of Ephermerl Eyes (depending on your World Tendency).
• Revive and get yourself killed in that particular World that you are working on until you reaches Pure Black World Tendency.
• Find and kill the Black Phantom NPC
  • World 1 - 1-1 Miralda, left of starting archstone.
  • World 2 - 2-2 Scirvir, jump down to reach him. If the fog in the tunnel is not activated you are screwed because you can't get to the Primeval Demon.
  • World 3 - 3-1 Rydell
  • World 4 - 4-1 Satsuki, start of 1-1.
  • World 5 - 5-2 Selen, near First Fog.
• Now, without returning to the Nexus, you must find your way to the Primeval Demon and kill it.
  • World 1 - 1-3 Just go through the gate and you should find it in a location over looking the 3 Red Eyes Knights.
  • World 2 - 2-1 At the bottom of the moving platform below the 2 Rock Throwers at the beginning on 2-1.
  • World 3 - 3-2 At the bottom of the swamp.
  • World 4 - 4-2 In the area behind the Dual Katana Skeleton in the hidden passage.
  • World 5 - 5-2 In the swamp near the slug nest.
• Once you had killed the Primeval Demon, return to the Nexus. You should be in White World Tendency.
• Kill 1 Boss in Soul Form (not recommended to do it in Body Form because you will lose World Tendency if you screwed up).
• Return to Nexus, you should be in Pure White World Tendency.

It kind of depends on what you have. I have a ring that makes it more likely for the enemy to drop stuff. Also, the higher your luck is, the more likely they are to drop items.

Cool, but is this a game where every half hour or so I will have found new, stronger weapons and/or armor? I like when games have a ton of drops.

not at all. weapons can be found on each level but there is focus on upgrading them to suit your character. So while you find many weapons, you're only going to upgrade one or two to actually use. I'm level 53 and only on my second set of armor (wizard's clothes -> leather). It's not like Diablo or Titan Quest where you get new armor pieces all the time, and need to switch them out in order to survive.