1. I can not update to the newest version or down load anything from a the App Store. When I do it fails and says I do not have permission to down load.
I agree with most of what has been said. The last thing to do is make sure to repair permissions if nothing else works. Applications -> Utilities -> Disk Utility has the function for that in its First Aid section.
2. When I conect a camera to upload picture, It will not recognize any of my cameras. They never show up in devices on Finder, Iphoto, Apperture nor in the Utilities.
This can be a sign of a camera not complying with normal USB standards, or one that doesn't understand Energy Saver. Make sure to connect it only when the machine is plugged into the wall, and you have not closed the lid to the display, or allowed it to fall asleep during the current boot of the machine.
If the camera uses digital media card, it is better to use a USB Digital Media Camera reader, or the built-in SD slot (only supports SD cards), if the machine has one, or the Express/34 slot with an media card adapter if the machine has one. This will save the camera battery life, and be more compatible with your machine, and is easier to shop for compatibility issues.
3. Anything that I try to airdrop from my wifes computer fails.
http://support.apple.com/kb/HT4783 may show that one machine is not configured or right or isn't compatible with it.
4. Someone told me to delete a file called "Library/prefrences/com.apple.finder.plist. I did not delete it becuase it does not exist!
Note, there are three libraries. Check in each. Boot hard drive -> Library, Boot hard drive -> users -> yourname -> Library, and Boot Hard drive -> System -> Library
5. The whole system crashes frequently when working with Aperture or I photo.
When you say crash, do you mean you just get a spinning colorwheel? Have you tried command-option-escape to see if you can force quit the application if that's what it is doing? Granted you'll lose any unsaved changes, but at least you'll get back access to your system to try and clear it up. If you are getting a four language screen asking you to restart your computer now, that's a different story, known as a kernel panic. Don't panic if you get one. Rather read up on those*:
http://www.macmaps.com/kernelpanic.html
If a spinning wheel won't let you get the force quit window, it may be you have too much data on your hard drive. Make sure it isn't over an arbitrary 85% full, where Mac OS X typically has always had a bit of a hiccup trying to deal with*:
http://www.macmaps.com/diskfull.html
Also make sure the versions of Aperture and iPhoto are known to be Lion compatible. Since you say you are using AirDrop, it is obvious you are also using Lion.
This link has been a mainstay for people to check for Lion compatibility.
6. The apple store replaced the mother board over a year ago. They have rebuilt my photo libraries. The trouble is that I am overseas where there is not an Istore just an IStudio 300 miles away and I dont think they know that much.
Always a problem when you travel. Contact http://www.macsales.com/ for a portable 2.5" external hard drive to use for backup of equivalent size to your existing drive. http://www.macmaps.com/backup.html explains your backup options. Once backed up, we can examine if you actually have a hardware or software problem if none of the above solutions seem to work.