1. yes
2. you still need to control forr fixed effect after first level differences, if you assume the treatment effect varies with time or group..
3. PSM assumes selection-on-observables while DD is for unobservables. Propensity score methods should be able to control the difference b/t treatment and control groups using all the variables that you have, and then you can reduce the dimensions to do whatever you wanna do, like matching, regression,etc; DD assume the treatment and control groups are very similar though you have unobservables that might bias your estimates. By differencing all the unobervables, DD is supposed to give you unbiased estimates. Read Almond and Chay's the cost of low birth weight, you will be able to know PSM a bit. For DD, read Card and Ashenfelter's (I forgot the title, but it was one of the labor econ lit)
4. pscore is a command in Stata, but for DD, you need to construct variables by yourself.
hope that helps