Answers:
1. To copy a DVD based game with a DVD writer.... I would suggest using Primo DVD.... Here is a step by step tutorial on how to make a working backup of a DVD based PS2 game using Primo....
Primo DVD Tutorial
2. To copy a CD based game with a CD writer.... I would suggest using Clone CD and a 100% compatible writer... If your burner is 100% compatible with Clone CD, you can make a 1:1 copy of it and not have to patch it....
Go here to see if your burner is compatible:
Clone CD supported writers!
Hope this info help you out....