The sksapps site is basically the go-to site for PS2 apps at this point I'm pretty sure.
As to how to do it yourself, as previously mentioned you'll need a Codebreaker or AR disc if you're starting from scratch. Alternatively, you can just mail a memory card to any of a host of people out there and they can do it for you, since once you have a FMCB card you can use it to make other ones, and it's ridiculously simple. No actual modding of any sort is done to the PS2, firmware or hardware. But it's good that you have a fat model - loading from the internal HD is far preferable to USB/Network loading (though I've heard a few people say network loading is okay. I've never used it myself)
Yeah wonderful as they complement each other, neither a softmod or modchip changes the BIOS.
Your console will in effect remain its original internal region. Like no menu option for Japanese on a US system. Homebrew is needed to get around that.