What do you think of this one?
http://bit.ly/vn1QGC
Do you think this could run Minecraft/Blockland/World of Warcraft no lag?
Yes, it could.
My CPU is the AMD Phenom II.
Also what do you mean by "With a lane below it free for the cooler?"
You won't experience any bottlenecks with it.
It's a dual-slot card, it takes the room of 2 normal expansion cards. One being the outputs and everything, while the other one is fan input/output.
What's that mean? As long as I don't try to overclock the GPU it'll be peachy?
Pretty much. This also means you cannot upgrade to a greater card without upgrading your processor.
How could I find out if either my router sends slow wifi or if my adapter is just slow?
I'm assuming you're using Windows.
Click your network AP's name in Network and Sharing Center. For XP: go into Network Connections, right-click your wireless device, and click "Status. Monitor the speed while you're doing your normal network stuff. If it goes down, it's likely your adapter. If not, it's your router/AP.
Do motherboards have a built in wi-fi receiver?
No. You'll have to buy one separate.
Wait, is this: "DirectX Card: ATI Mobility Radeon HD 6370" the graphics card. I don't really know. I think I was looking at the driver.
Yes. That is your graphics controller.