To what extent can phase transitions be made rigorous? It seems a lot of physical intuition in statistical mechanics, for example phase transitions, critical temperature, scaling hypothesis, renormalization group methods etc. should have a purely mathematical formulation; my question is: to what extent can this be done? Can we prove statements from a standard textbook (say Statistical mechanics by Huang) in a mathematically rigorous way?
A more specific example: it is well know that the 2D ising model with no external magnetic field has a 2nd order phase transition, can this be proven rigorously?
 A: Yes, a lot can be proven rigorously, at least for lattice systems.
There are entire books on this topic. Here are a few (the first one can be legally downloaded for free):

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*Friedli and Velenik, Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction

*Georgii, Gibbs Measures and Phase Transitions

*Ruelle, Statistical Mechanics: Rigorous Results

*Simon, The Statistical Mechanics of Lattice Gases

*Prum, Stochastic Processes on a Lattice and Gibbs Measures

*Presutti, Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics

*Fernández, Fröhlich and Sokal, Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory

*Israel, Convexity in the Theory of Lattice Gases

*Ellis, Entropy, Large Deviations, and Statistical Mechanics

*etc.

Concerning the specific list of topics you mention, let me mention that rigorous results about the critical behavior remain scarce. Exceptions are:

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*in some planar models (in particular, the Ising model), a lot of progress has been made in the last 15 years, since the introduction of the Schramm-Löwner evolution made it possible to prove conformal invariance; (one example)

*in some cases, it is possible to implement rigorous renormalization group methods to obtain information about the critical behavior of nonintegrable perturbations of integrable models; (one example)

*it is also possible to implement rigorously the renormalization group method in sufficiently high dimensions; (one example)

*in sufficiently high dimensions (at the very least, above the upper critical dimension), an alternative way of obtaining detailed information about the critical behavior of a variety of models (including Ising) is the lace expansion. (one example)

As to your question about the order of the phase transition in the Ising model, I don't think any of the books I mention above proves the fact that the latter is of second order. In dimension $2$, this follows, for instance, from Onsager's exact expression for the free energy. In higher dimensions, continuity of the magnetization at $\beta_{\rm c}$ is proved in this paper.
