Is such a stopping and starting again equivalent to a entering a regular "sleep" mode? Or do the RAM contents get lost and it's effectively a reboot?
Stop/Start is the same as shutdown/startup.
So to answer your question, running processes need to exit gracefully on shutdown lest they be killed. Then on start, you either need to manually start them or start them automatically via your distro's mechanism (upstart, sysvinit, etc.).