to download the source code you need an SVN Client (like TortoiseSVN(most preferred) or RapidSVN)
and to answer your questions
1. it's at http://code.google.com/p/dolphin-emu/source/browse/trunk
and if you want to download it you use http://dolphin-emu.googlecode.com/svn/trunk/ in your SVN client
2. to patch the sources (with TortoiseSVN) you just right click where you download them and click "TortoiseSVN->Apply Patch" you look for the patch file then you right click in the window to the left and click "Patch All"
3. you can use Visual Studio, Visual C++ Express or MingW
and to answer your questions

1. it's at http://code.google.com/p/dolphin-emu/source/browse/trunk
and if you want to download it you use http://dolphin-emu.googlecode.com/svn/trunk/ in your SVN client

2. to patch the sources (with TortoiseSVN) you just right click where you download them and click "TortoiseSVN->Apply Patch" you look for the patch file then you right click in the window to the left and click "Patch All"

3. you can use Visual Studio, Visual C++ Express or MingW