How many different ways a DNA molecule could be altered?

It depends what you mean. If you just mean in code it can be altered in 3^x ways if x is the number of base pairs. The process of doing this is through mutation of which there are many mechanisms and causes. If you mean in other properties it is possible to make left-handed DNA (though not found in nature), you can substitute certain molecules like putting in sulfur instead of phosphorous in the backbone (certain bacteria can do this for a small fraction of the phosphorus in their phosphate backbones). Theirs also modified bases that take the place of normal nucleotides and you can add things to DNA such as methylating bases or binding proteins.