What you want for that is rsync. It was written just for what you want to do. It syncs the contents of one folder to another. It can be configured to only transfer changed files, so the first run will be long, and copy every file in the source folder. Subsequent runs will be quicker, depending on how many files have been changed. Rsync is usually run through a crontab, at a time when nothing else is going on. It can log everything if you want, or whatever you want. In a terminal, run
and you will get the parameters to use, or perhaps easier, there are several websites that explain rsync. AFAIK it only syncs one source folder per run, so you may need to run it multiple times to get all of them synced, but one run can do it if they are are all under one upper level folder. Study the documentation and come back if you need more help.
As for creating an empty file for the clamav log, that is necessary only if you use the >> redirector. If you use only > the file will be created for you. Don't ask me why, that's just the way it is. For knowing where to look, run clamav and tell it to report only errors. I'm not really familiar with it, because I don't bother with a virus checker. It just slows things down and provides no real advantages unless you're running Windows.