After it has finished downloading, does installing Yosemite require an internet connection to install?
No. All of the information has been downloaded therefore it does not need an Internet connection to install. Just make sure you start the install while on wifi because the installer asks a few questions and it may need wifi to process it. Once answering the questions and it starts the actual install you may take the computer away from wifi
No, there is a difference between download and install. Download is to get the files from the Internet, and install is applying the downloaded data.
However on most OS installations, an internet connection is recommended (Sometimes necessary). Sometimes, the installation goes wrong and in that fact, you might need Internet Recovery to fix it.