<p>The published code for Palm OS applications doesn't include standard headers, but instead uses Palm OS APIs for things like StrCopy and MemMove.  Can I use standard headers and functions or do I need to convert my code to use the Palm OS versions?</p>

It depends on the compiler and what version of the tools you're using.  In general, if you can use Palm OS APIs, you'll probably work better on the platform, but both CodeWarrior and prc-tools had some library functions implemented.
In CW for Palm OS V9, there's a pretty full version of the Metrowerks Standard Library (MSL).  Most of MSL C++ is there giving you access to STL and other constructs.  On the C front, you've got most of the standard C headers for memory management and string functions.  However, it omits implementations for locales, math, signals, and standard I/O.
To use the library, you need to make sure you link with the MSL C/C++ libraries.  This can be set when you create your project in the wizard or added later by modifying the access paths and adding the appropriate static libraries for your targets.