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This is the index to most of the java tutorials. You can find many things here:
The Really Big Index

Why AWT? That is the old technology. Most GUI code is now written using Swing.

I have no ideas for projects for you.

Applets need permission to be able to write files.
See the tutorial for how to write an applet:
Lesson: Java Applets (The Java™ Tutorials > Deployment)

If its your computer, you should be able to store the images where ever you want them.