The difficulty is that these logical arguments depend on the rationality of the actors in the economy. And while Mises acknowledges that the real reasons for a person's actions may not even be known to one's self, more recent experimentation seems to show that at least in some situations, people really do not behave rationally.
Anyway, if you're concerned about a lack of mathematical rigor, but are interested in Austrian ideas, you should look at the Chicago school, with luminaries such as Milton Friedman. The Chicagoans started from Austrian roots, but if anything, their attitude toward mathematical models is the opposite of the Austrians: they virtually invented much of the mathematical techniques that are employed by modern economists.
I'm also not sure what you mean when you distinguish logical from mathematical models. Could you explain?
Austrians are pretty much hard money. There are some interesting comments made by rothbards and Friedman about each other ot illustrate the split.
The schools reach similar conclusions on a number of issues, though.
You already accepted they are separate, I'm just trying to add flavor here, not win a point.
Praxeology simply claims that you can't predict the economy based on experiments where you stick 30 college students into some iterated prisoner's dilemma scenario.
Austrians strongly favor models, math and science. They just want robust models which are not sensitive to parameters, and econometrics of revealed preferences rather than highly controlled experiments.