No, I don't see any way to remove them, regardless of what I click on. I was browsing through the list of available widgets and didn't realize that I'd installed the same ones more than once in some cases (like the Analog Clock entries you see), and I have not found a way to get rid of the dupes yet.
As for the Application Launcher, I just messed up this laptop pretty bad. LOL
I added it back to the panel as a first step. Then, when I went to move it to where I wanted it (on the left side of the bottom panel), I accidentally "shrank" the part that has the clock, volume, etc. I couldn't figure out a way to get it back the way it was when I was still in "Panel Settings" to move it. So, I decided to "get smart" and powered down the PC (long hold of on/off button, thinking that my changes where I messed up would not have been saved yet). That was an even bigger mistake, as after rebooting, it was only partially visible and I couldn't get to what was being shown at all. Grrrrrr.
It's a darn good thing I backed up this laptop earlier. I made a full disk image backup using ddrescue from a Live Mepis 8.5 DVD (using the 8.5.03=rel1 USB ISO I used to install Mepis), before and after I changed around his Windows config and partitioned the drive). I also made a separate image file of sda7 (the partition I had Mepis 8.5 installed on) after I thought I was mostly finished with changes. So, I just booted back into the Live DVD and restored the partition from the backup image file (I've got it on an external drive).
Here's all I needed to do to restore the partition (copying the sda7.img file I made earlier from a USB attached drive using ddrescue). Please ignore my typos changing folders. LOL I just plugged in the USB drive, clicked on it to open Dolphin and mount it, and used ddrescue to restore the image file I had of /sda7 (saving time compared to restoring the entire drive).
Click on the image for a Larger Version:
[url=http://www.pbase.com/image/124178083/original.jpg]

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So, I'm back to where I was before I messed it up. When I finish with this laptop and go over the use of what I've changed with it's owner, I'll make sure to suggest that he does *not* attempt to change the Panel, add more Widgets to the Desktop, etc., without a really good backup. It's too easy to mess something up. I'll make sure to leave the panels and desktops locked. :-)
P.S.
IOW, it's going to look like this again, and I'll need to make the changes to the preferences file, Synaptic, etc. one more time (since those were made after my backup of the partition that I just restored from an image file of it).
Click for larger version:
[url=http://www.pbase.com/image/124172520/original.jpg]

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IOW, you only need one command (as root) to restore an image file of a partition from a mounted USB attached drive after booting into Mepis from a Live DVD or CD). For example, I'm restoring an image file named sda7.img on /media/disk (an external USB attached Drive I plugged in) in this case, restoring it to /dev/sda7 on the laptop's hard drive.
su
ddrescue /media/disk/sda7.img /dev/sda7
It's just as easy to backup a partition or drive using ddrescue (only one simple command as root). Whew (I'm glad I backed it up earlier this morning)