The plugin components, AUs, are placed in either /Library/Audio/Plug-Ins/Components or ~/Library/Audio/Plug-Ins/Components. The ~, or tilde, stands for the home directory.
The reason there are two possible locations is that the /Library level is global for all users, while the ~/Library is local for the currently running user. In most cases it’s best to install the plugins globally. Hence most installers will place the AU plugins into the ~/Library location. It’s good to know this location in case you have to install the plugin by hand, or remove it of some reason. It’s also good to maybe do backups of this directory from time to time, even if I personally have backups of all the install program for specific plugins, as the plugins might install patches, documentation and other data files.
Logic will actually run a program called AUVal to validate any AU plugins installed, it will make sure that the plugin could properly run, as a lot of supposedly DAW related problems could be a result from a badly working plugin. AUval is a command line tool, so if you are interested, you could also run this by hand. Ableton Live is not running AUVal.
As for where the patches are possibly installed, this is not a standard. Usually you could find them installed in the /Library/Application Support/, somewhere inside there, or then use Spotlight with a known patch name to find where they were located. Fortunately, most plugins save the patch directory as a preference, so next time you open up patch files, you could quickly find the rest. I’ve even seen some installers placing documentation inside the ~/Documents folder, not in the /Documentation folder which I would think is the most obvious place for system-wide documentation. Or, Ableton Live has its PDF documentation file inside the main directory that you download when installing Live.