I have not automated a Java web app before. I am using vbscript in test complete to complete this task. Two questions here: 1. In general, Is it best to use the name mapping or the full name? ...
>>1. In general, Is it best to use the name mapping or the full name?
It depends on your skills and background. Some TC users prefer using NameMapping and the Alias tree.
As for handling dynamic objects (whose identification parameters are changed from run to run), watch this video - it shows TestComplete's NameMapping features that you can use to create a reliable test.