Upgrading my CPU, need advice

1: There are other limits than just the CPU socket. One example is the chipset. Most likely, it's running a pretty old chipset, which means the flow of data in your computer is probably going to be significantly slower than if you got a newer motherboard with an advanced chipset.
Another limit on an older motherboard is the lack of SATA 3 which has a much faster transfer rate than SATA 2 (double actually).
Yet another limit is that PCI-E is significantly slower on older motherboards (since of course, a new revision has come out).

2: Do not touch the aluminum plate on the top of it. This will reduce the effect that the thermal paste has on cooling your CPU. If you aren't using a new CPU, you'll have to remove the current thermal paste that's on both your CPU and you CPU heatsink.
When you go to install a CPU heatsink (like the CoolerMaster Hyper 212 Evo), screw in the screws about one turn on each screw so that you don't unbalance the thermal paste, and screw the screws in a diagonal fashion (A summary of what I'm explaining can be found here at about 15:30 into the video: http://www.youtube.com/watch?v=d_56kyib-Ls ).
And make sure that the triangles line up when you are mounting the CPU. If they don't, then it won't work, and will most likely damage the CPU.

3: Yes, it's almost a given that your best option is just buying new and updated components.

Linked below are the parts of my custom computer:
Motherboard: http://www.newegg.com/Product/Product.aspx?Item=N82E16813131757&Tpk=asus%20m5a97evo
CPU: http://www.newegg.com/Product/Product.aspx?Item=N82E16819106010&Tpk=amd%20fx%206200
Heatsink: http://www.newegg.com/Product/Product.aspx?Item=N82E16835103099
RAM: http://www.newegg.com/Product/Product.aspx?Item=N82E16820145345
The total price for that is: $364

If you think that that is too high of a price, you could opt out of the aftermarket heatsink, and get a slower CPU, and you could go with 4 GB of RAM instead of 8 GB (which I wouldn't recommend). So without the heatsink, slower CPU (linked below), and 4 GB of RAM (also linked below), your price is $250.
CPU: http://www.newegg.com/Product/Product.aspx?Item=N82E16819103996
RAM: http://www.newegg.com/Product/Product.aspx?Item=N82E16820145346
Honestly, this $250 is a great entry level gaming computer. All you need to do is pair this up with a decent graphics card (which you'll probably need), and maybe even a better power supply, and you should be set!

Just a personal note: I am willing to help you through this process step by step if you want. Honestly, I like doing this kind of thing, and since I'm already on summer break, I've got nothing better to do (other than Minecraft....). You can contact me at my email address listed on my profile if you would like assistance.