Is there a way to have Table of Contents display with certain folders open as the default view for the user when he first opens the web-based help?
The short answer is NO.
The long answer is that you may have one folder open by default. The way you do it is to configure TOC synchronization, then ensure your default topic is inside the folder you want open.
When the help loads up, the TOC synchronizes to the topic and expands the folder to show the TOC node.