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I am making a game engine that will run in an applet, that would need to be able to do things like Perspective transform (for Mode 7), basic transforms (drawing an image at custom corner points), etc, and if possible could either of those be embedded directly in my applet without needing the end user to install them?
Java2D is included in the JVM by default – there are no additional requirements to use it, even from an applet. You can perform any affine transformation with Java2D. This includes a perspective transformation, or translation to render images in various ways.
Here’s an example of using Graphics2D with an applet:
http://download.oracle.com/javase/1.4.2/docs/guide/2d/spec/j2d-awt.html
Java Advanced Imaging is not part of the JVM, but you can bundle it with your project AFAIK: http://java.net/projects/jai/