Do different fat contents (skim, 2%, whole, etc) affect rate or intensity of milk spoilage?
No, there is no reason they should. 

Bacteria feed on carbohydrates, not on fats. (This is why oil doesn't spoil outside of the fridge - it is pure fat). So it is the amount of milk sugar which is important for the bacteria, and it is the same regardless of the fat content. Also, the spoiled milk is not more or less sour at the end. 

The other important factors are initial bacteria count and storage temperature. You can't do anything about the bacteria count, and can't know what it is. The storage temperature is, of course, your fridge temperature, which should be in the small range of 0-4°C. 

The other factor is the type of pasteurization. Traditional pasteurization makes a milk with a shelf life of 7 days, ESL gives it a shelf life of 21 days, and UHT milk can last for months, uncooled. Once opened, each of these types spoils within 3-4 days. And then, normally pasteurized milk goes really sour, partly even separating into curdles and whey. ESL and UHT milks get slightly more bitter, but not much, and they stay liquid and don't change their odour or colour. Unless you pay lots of attention to the taste, you can't tell when ESL milk is spoiled. But this happens the same way no matter the fat content.