When creating the local/props.conf and local/transforms.conf, do I need to copy the entire default/props.conf and default/transforms.conf files into local/ or do I simply start with blank files and add the sections I need?
No you do not, only the settings you want to overwrite or change from the default.
There is a default stanza that all props inherit settings from, you can find that in default.
If you run btool --debug against your sourcetype, you will be able see all the settings that are applied and inherited from the default, along with which files Splunk is reading the configuration from. Good to always check!