First, what happens when you press the windows key on the keyboard.
On to the next step
Press CTRL+ESC. If your taskbar magically reappears, then it's probably simply been resized too small, repositioned somewhere you didn't expect it. If CTRL+ESC didn't make it come back, then the program that displays the task bar most likely isn't running. That program is often called the "Windows Shell", but it's really "explorer.exe". It's the program that not only displays the Taskbar, but also responds to the CTRL+ESC sequence.
The workaround: press CTRL+ALT+DEL to fire up the task manager. Now on the File menu, select New Task (Run...). Type in "explorer" and press OK. That should restart explorer and re-display your taskbar. Run an up-to-date anti-virus and anti-spyware scan. Both viruses and spyware are known to occasionally interfere with the Windows Shell.
Then, I would run the system file checker: What is the System File Checker, and how do I run it?. That will check for missing or damaged files in Windows, and repair or replace those that it can. To run the system file checker press Start, Run, and then type in the below
This causes SFC to scan your system immediately. SFC can take a few minutes to run. As I said above have your installation CD or equivalent available in case SFC needs to replace a damaged file.
While it's not documented anywhere, if SFC replaces any system files I'd reboot. I just like to make absolutely certain that the file replacement actually takes effect.
Finally, you may be faced with re-installing Windows if none of the above work.
On the running of programs from the desktop, i would check th see what the file size is on the program. You may be just clicking a shortcut and with the problem of the taskbar you have, the computer may not be able to find the program. One thing i do with all my computers is make a folder on the "C" drive called Downloads and i put all downloads from the internet there first, then move them to where i want them later on.