Text-Classification-Problem, what is the right approach?
I think perhaps the first thing to decide that will help clarify some of your other questions is whether you want to perform binary classification or multi-class classification. If you're interested in classifying each instance in your dataset into more than one class, then this brings up a set of new concerns regarding setting up your data set, the experiments you want to run, and how you plan to evaluate your classifier(s). My hunch is that you could formulate your task as a binary one where you train and test one classifier for each class you want to predict, and simply set up the data matrix so that there are two classes to predict - (1) the one you're interested in classifying and (2) everything else.

In that case, instead of your training set looking like this (where each row is a document and columns 1-3 contain features for that document, and the class column is the class to be predicted): 

1           2           3           class
feature1    feature2    feature3    politics
feature1    feature2    feature3    law
feature1    feature2    feature3    president
feature1    feature2    feature3    politics


it would look like the following in the case where you're interested in detecting the politics class against everything else:

1           2           3           class
feature1    feature2    feature3    politics
feature1    feature2    feature3    non-politics
feature1    feature2    feature3    non-politics
feature1    feature2    feature3    politics


You would need to do this process for each class you're interested in predicting, and then train and test one classifier per class and evaluate each classifier according to your chosen metrics (usually accuracy, precision, or recall or some variation thereof).

As far as choosing features, this requires quite a bit of thinking. Features can be highly dependent on the type of text you're trying to classify, so be sure to explore your dataset and get a sense for how people are writing in each domain. Qualitative investigation isn't enough to decide once and for all what are good features, but it is a good way to get ideas. Also, look into TF-IDF weighting of terms instead of just using their frequency within each instance of your dataset. This will help you pick up on (a) terms that are prevalent within a document (and possibly a target class) and (b) terms that distinguish a given document from other documents. I hope this helps a little.