Q: How do you get the date and time to display when you show the video you have taken?
The Date and time is set but does not show up when I play back the videos I have shot. — Robert Davis
A: "The data code (date/time or various settings when recorded) are not displayed while recording. However, they are recorded automatically onto tape. To display the data code, press DATA CODE on the Remote Commander during playback (p.47)." —DCRTRV350.pdf manual
The data code information acompanies the data stream when copying with the iLink (FireWire) cable.
Power up in VCR mode and use the infrared remote (RMT-814) key "Data Code". The first press shows date and time, the second press shows f-stop, white balance, shutter speed, gain and other parameters of the recording. The third press returns to hiding the data code information.
http://www.docs.sony.com/release/DCRTRV350.pdf <— get the manual (from http://esupport.sony.com/US/perl/model-home.pl?mdl=DCR-TRV350&region_id=1)
Read Page 47 - "Playing back a tape" - "to display the screen indicators".

Q: How do you get those giant cats in the sky? — Alyn Smythe
A: 3D, i iz cheezburger and pickle in side - Mickey Mouse, Minnie Mouse and Wizard flee the Giant Cat, Aghh, Running, Main Street U.S.A., Disneyland®, Anaheim, California, 2008.11.14 13:21 contains a background which replaced the blue sky using the "Memory Mix" function that combines static images from the memoy stick onto your recording.
You can download many backgrounds, overlays, frames, watermarks and mattes at trv351 and read my description of their use.
The manual has detailed information on page 140 under "Giving Images More Impact".
I had the cat image loaded on the memory stick. Before recording, I engaged the "Memory Mix" function behind the flip out LCD display (in the upper right corner), then picked which image to use with the adjacent memory buttons (+ & -). When I had the cat previewed in the corner of the display, I rotated the thumbwheel to C.CHROMA and depressed it. Now anything blue, like the sky, that was imaged in the lens was replaced by the image on the memory stick - the giant cat.
This doesn't always work. Sometimes there are clouds in the sky. In this case the sky becomes less blue at the horizon, giving the image a brownish gray where cat should have been.
Frames, watermarks and titles can be done the other way round, where the blue or white part of the memory image is replaced by the image in the lens - M.CHROMA or M.LUMA.
The last thumbwheel selection fades from the selected memory mix image to the lens, great for establishment shots with M.OVERLAP.

The famous Giant CAT! image is a stereograph (cross-eye freeviewing, no special glasses required.) I had two cameras calibrated to point to a very far spot on the horizon. I set the C.CHROMA memory mix to the same giant cat in both cameras, rolled tape, and used the photo button on the remote to synchronize the snapshots. I had five minutes to load the settings and roll the tape in both cameras, or they would automatically shut down to prevent the spinning record head from burning the tape. Fortunately, the image selection is retained when switching modes (including off and charge), so you can take your time and pick the image first in Memory mode, then switch to Camera mode, press Memory Mix and your selected image is right there, ready to depress the thumbwheel mix type. 

What made it famous? As seen on BBC4, on Mickey's 80th birthday, when they closed the evening news broadcast with a human interest piece "Walt Disney built a better Mouse" they rolled the credits over this picture.