Again, Sharon, thanks for your time.
With regard to the My Computer icon; yes, I do use the Classic style. I am
able to drag the My Computer Icon while in highest level of Explore into the
Task Bar/Start Menu, and it works fine. But the icon disappears each time I
run a Clean Cache 3.0 program.
With regard to the Recycle Bin folder, it does not appear anywhere in
Explore, no matter what level I look at. All I see is the RECYCLER folder.
Could have I hidden it with a previous Registry change?
Hi, again. The first thing that popped into mind after reading your first
paragraph was "then don't do that."

Hard to keep something that a
program insists on removing? Any place in that program's settings to define
exclusions?
For the second: I don't know what registry changes you did but yes, a
registry change could do that. If I collapse everything except the main
entries under My Desktop, I see My Documents, My Computer, My Network
Places, Recycle Bin.... Wait a minute. I *do* keep the Recycle Bin on my
desktop and you said that you've hidden it. Perhaps that is why it is not
appearing for you? You could try putting it back on the desktop. Create
your shortcut. Then hiding it again. Might work, might not.
Also, MVP Kelly Theriot has some tweaks for the Recycle Bin on her website.
The third one I've listed may be particularly useful to you at the moment.
It might also require reversing what you did before and then applying her
tweak but don't know for sure.
http://www.kellys-korner-xp.com/xp_tweaks.htm
Replace/Repair Recycle Bin
Line 64: right hand side
Rename/Remove Recycle Bin
Line 115: left hand side
Hide Recycle bin on Desktop - Not from Explorer
Line 199: right hand side
It sounds like you are accustomed to tweaking Windows so will leave out the
"protect your backside" talk. But have included the Replace/Repair tweak of
hers "just in case."
Another thought:
Possible alternative that may keep the shortcuts for these items in place:
Put them in your Quick Launch bar. They would still be handy and only one
click away instead of two. Or at least use QL (or another location) to
store copies of these so that you don't have to recreate them every time
they disappear?