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How do I customize the format of the time string?

The widget and template tag allow you specify a time format directly. The default value for the time format, and the one used by the display of the blog time in the static admin widget, can be overridden by adding a filter to 'blog_time_format' and returning the desired time format. See http://php.net/date for more information regarding the time format.

NOTE: The time string is currently only configurable for the static clock and the widget, not the dynamic admin toolbar clock enabled by default.

Why is the time not changing in the sidebar widget?

This plugin does not (yet) provide an active clock that continues to update to reflect the current time as time passes for the sidebar widget. It merely displays the current time, according to your server, at the time the page was created and sent to your browser. You can click on the time itself to see it dynamically refresh (without a page reload) to the current time. Or if the page gets manually reloaded you'll see a new current time. The dynamic clock is currently only available to the admin toolbar widget.

The time matches my computer's time; how do I know this thing is working?

Your machine may well be synced with the server's clock. One test you can perform is to change the blog's time zone (under Settings -> General). The blog's time will then be set to a different hour, which should then be reflected by the widget.

How do I go back to having the static timestamp as opposed to the dynamic clock?

See the Filters section for the c2c_blog_time_active_clock filter, which includes an example line of code you'll need to add to your theme.