Yes, it is possible to model chemical reactions with QM. In quantum chemistry, one often solves for the Schrodinger equation of the molecular Hamiltonian assuming the Born-Oppenheimer approximation. Within this approximation, the energy is a function of the coordinates of the atomic nuclei. Thus, to model a reaction you can follow the lowest energy path from the atomic coordinates of the reactants to those of the products.
Now, how complex can the reactions be and still be computationally feasible to model? That depends on two things: (1) how much accuracy do you need? (2) what kind of effects are involved in the reaction?
For example, with modern computers and using Density Functional Theory (DFT), it is perfectly feasible to model the methylation of benzene (or even much larger systems), but I wouldn't expect an accuracy better than around 4 kcal/mol in the energies. However, I do not know of any method that is able to give an accurate description of the dissociation of the Cr$_2$ dimer, even though larger systems can be modelled accurately with a variety of methods. The reason for this is that the effects of electronic correlation are very strong in Cr$_2$, and these effects are inherently difficult and computationally costly to take into account.
If you want the best possible result (i.e. taking into account all of the correlation effects), you need to use a method called Full Configuration Interaction (FCI). In FCI, you minimize the energy of a wavefunction which is a linear combination of all the possible configurations (Slater determinants) that the system can take. This is the most general wavefunction possible and thus, by the variational principle, when you minimize its energy you get the exact numerical result for the Schrodinger equation for a given basis set (i.e. the functions that you use to represent your orbitals). However, the time complexity of this method is roughly $O(M!)$, where $M$ is the number of basis functions, and the result is really exact only when $M$ tends to infinty. In practice, $M$ only needs to be very large to converge but this is still so costly than we can only afford to do FCI for systems with no more than around 10 electrons. This is why there are so many approximation methods in quantum chemistry; so that one can use a method appropriate for size of the system and the accuracy desired. DFT is extremely popular because it is relatively cheap and accurate (formally, $O(M^4)$ and average errors of about 4 kcal/mol in energies as noted above). Other good methods that are in between FCI and DFT in accuracy and cost are coupled cluster methods ($\sim O(M^7)$) and CASPT2 (combinatorial cost), which is a kind of FCI in a subset of electrons and orbitals and adds a second order perturbation theory correction on top of that.
With respect to examples of reactions, you should probably check for the Woodward-Hoffman rules (http://en.wikipedia.org/wiki/Woodward%E2%80%93Hoffmann_rules). These are some simple rules based on QM and molecular orbital theory which can predict the outcome of a great deal of reactions that could not be explained in any other way.
