Issues regarding website with old code

You don't need to download all 2.x and 3.x versions and apply all system folders (or application/config). You can grab the latest. So no need to go 3.0.1 to 3.0.2 etc, you can jump to the latest. But replace all files mentioned with the latest version.

But you should read all upgrade pages, as they contain information about deprecated functions.

Personally I had a development with PHP 5 the last time I upgraded and upgraded to the latest 2.x before upgrading into 3.x. So that I knew where I broke stuff. :-)