You don't know.  As much as you can, try to adapt your application to
work correctly on a variety of screen sizes.  If you can't do so, then
you'll need to check the screen size and put a border around the game on
larger screens.

No.  The correct approach to writing Java code to run on mobile phones
is midlets.  Midlets don't really track the same Java platform version
numbers as the J2SE.  They are loosely based on Java 1.1, but the entire
standard library is replaced by the CLDC and MIDP, which provide some of
the basic java.* packages as well as midlet-specific user interface
classes.  You won't be able to use AWT or Swing, or most of the Java 1.2
standard API.  Also, midlets based on CLDC 1.0 are forbidden from using
floating-point numbers.

PDAs are more complicated.  Some of them implement CLDC/MIDP, while
others implement can do CDC and PersonalJava and even the full J2SE.  It
depends heavily on which device you are using.

Chat isn't really a resource-intensive task, so I'm sure that just about
anything would be fast enough to run a chat application.  That said, the
text entry mechanisms on a mobile phone aren't really good enough for
chat, though admittedly that doesn't stop 13-year-olds from sending text
messages all day long with them.