I assume that the navigation in TCH knows the absolutely exact time and continuously keeps updating it. Is there any way to see this exact time?

I don't think there is a way to see the current time on the navigation screen. The approximate arrival time to each destination will show when you are in guidance mode. I believe this is all you will see in regards to time on the navigation screen.

It keeps track of UTC not local time. There is a diagnostic screen that exposes it. I think.

To get to the diag screen, you'd have to have a 5.1 dvd, wouldn't you?

You might not need the 5.1 version...the following might work with 6.1 but I haven't tried it.

I think there is another diagnostic screen that can be accessed by toggling the light switch from off to parking light mode several times while holding the "info" button on the Nav unit. This all has to be done with the car in "on" mode (push the power button twice WITHOUT your foot on the brake).

I am almost sure that there is a nav screen page in there that shows the time of day (GMT or UTC). To get out of this mode you can power the car off then back on. The override menu on the 5.1 firmware also has a screen that shows the time of day.

Just be careful and don't mess around in these menus too much! There might even be a way to see your speed (as measured by the GPS) on these screens, but I don't know how you can see this screen while the car is in motion). Perhaps you can start the car in this mode and then drive off without leaving the diagnostic mode?

Read this entire post on accessing the diagnostic screens:
http://greenhybrid.com/discuss/showthread.php?t=11376
This works for both the version 5 and 6 DVDs.

One of the GPS diag screens does have the time.

Good luck.