I have a new iMac, Intel Duo Core, OS 10.6.2.
Two Questions:
1) Somehow I’ve managed to ask for my password every time I try to move something (pix, etc.) to the trash. I don’t want to have to do this each time. I’ve checked passwords in Help and Spotlight and can’t seem to find how to change this. How can I just drag the items to trash without having to put in my password?
2) I want to eliminate much of my “images”—10,000—before using Time Machine. Many images are ones that came with my new iMac—maps; icons, printers, etc. I’m not going to use them, but I don’t know if some of the icons—shapes, full bleed, etc.—are needed for some procedure. Also, if there are dupe photos, I’m assuming I need to keep only one copy. Thanks.

I am not sure about the images question.  how did you search and find 10K images?  Where are most located?  I dunno..

But, number one is definitely permissions related. Meaning that within your home directory, the other subdirectories (Pictures/Movies etc) do not have the correct rights for you to delete the files you wish.  You will need to verify permissions on one folder at a time.  You could do this ‘wholesale’ via the terminal; but by doing it manually you will learn a bit more and it is safer.

Right click or control click on the Pictures folder beneath your home folder.

YourUserName/Pictures

Choose GET INFO. 
At bottom of the window that pops up,  You should notice a section that is labeled Sharing & Permissions.
In that list should be your user’s name with Read&Write; Privilege.  Don’t worry about what else is in the box, just your user name.

You might have to click on the Gold lock to allow changes. 

IF your name is listed, make sure that Read & Write is also listed. If not, make the change so that Read & Write is to the right of your user name entry.

DO NOT DELETE any other name unless you KNOW with 0 percent accuracy that you don’t need that entry.  OTherwise, you make make things a whole lot worse.  (For this folder anyway)

When you are set, Click on the Gold lock to ‘lock’ the changes.
CLose the get into box.

Open the very same Get Info box and verify the changes are still listed there.  If not,  Remove your name and immediately re-add your name with the correct permissions.  CLose the get info box.

That should do it.  Repeat the process for the other folders in question.