Problems with Representative Sampling I am curious: what exactly are the problems caused by using representative sampling rather than random sampling whenever creating a subsample of large dataset for nonexperimental analysis. Further, how effective is propensity score matching whenever one wants to create balanced comparison groups for non-experimental analyses?
To answer your first question: it depends on the subsample that you want to use. 

A representative or stratified sample is constructed by dividing the population of interest into non-overlapping subsets, drawing a random sample from each subset, and then computing weights to adjust for the fact that not all elements from the sample had the same probability of being selected from the population. 

The advantage of using a representative or stratified sample is that you can use information that you have about the population to construct your sample and so have more reliable estimates of statistics computed for the population of interest. The disadvantage is that the weights that you computed are right for the stratified sample that you constructed but if you want to explore characteristics that differ between the strata then these weights might very well be the wrong weights. Your estimates would be biased and you probably wouldn't be able to adjust for this bias.

As for propensity score matching. You can only match on what you can see. You'll still have to deal with the problem of omitted variable bias. It's possible that propensity score matching increases rather than decreases the bias. How effective it is depends on what assumptions you make and on whether these assumptions hold.