What are other synagogues doing regarding reserving their facility?

Debbie (calabasas?),
Due to liability issues, we don't rent our facilities to outside entities.
I'm not sure how adding the rental calendar would resolve admin issues. Unless your calendar program has a function to prevent double-booking it's nothing more than informational and still requires admin oversight. Outlook (not the web app) has a function to allocate meeting rooms which could fit your needs but I don't believe there is a function to incorporate it (as a feed) into dpcalendar (that's a global issue with freeware). There are a number of web-based calendar programs that can do what you want (I think calendar wiz is one) but, like outlook, they are not free.

I think that further explaining your goals and objectives (what is your optimal end-point) might help.

IMNSHO, that misses the point altogether. Somebody needs to be in a position to say "No" when there's an attempt to double-book. Not only that, but I can easily imagine circumstances wherein one potential booking might be given preference over another, or there might be a reason to reject a booking outright. If the web site says the facility is available, but there's a reason not to rent it at that particular time, you might leave yourselves open to an argument.