The terms generated by the RG will all respect the symmetries of the microscopic action (though one have to be careful with anomalies). That's why people tend to directly write a low energy effective action, and do not bother calculating the RG flow of the parameters.
However, this implies that you do not know how the effective parameters relates to the microscopic ones (this information can be interesting depending on the problem). For example, if one studies the Ising model (say, on a cubic lattice), and is just interested in the long wavelength physics, one can do calculation starting with a low energy effective action, that is, a scalar $\phi^4$ field theory. But then, one cannot know how to related the real correlation length, or the real magnetization, to the effective ones computed from the effective theory. Neither can one compute the critical temperature of the model. (The only information that one can infer from the effective theory, if we don't know how to relate the effective parameters to the microscopic ones, is the universal features close to a 2nd order phase transition.)
But, at least in principle, one could do a RG calculation to relate the microscopic physics to the macroscale. In fact, in the example above, the perturbative Wilsonian RG fails (because the problem is strongly coupled), but one can use non-perturbative approximations to do exactly that.
EDIT
Concerning the additional questions.
1) Generically$^*$, all terms are generated (this is easy to see, if one draws the diagrams that can contribute for a given term). However, their value will depend on where the flow is going. If it goes toward the Gaussian fixed point, all terms but the quadratic ones will be small, of order $(\Lambda/\Lambda_0)^\Delta$, where $\Delta$ is the canonical dimension of the term. If it flows somewhere else, all terms can be large.
2) The low energy action can have additional (emergent) symmetries, but not less symmetry$^{**}$. For instance, at a critical point, a lattice model will have a conformal invariance at long wavelength. Or in the context of the Mott transition in the Bose-Hubbard model, the special particle-hole symmetric point have an emergent relativistic symmetry (whereas the microscopic model only has a Galilean invariance).
3) To me, the best way to understand the flow of the effective potential is to learn about the non-perturbative/functional version of the RG, where one write down a flow equation for the full potential, and one is able to treat all couplings (even the ``non-renormalizable'' ones). For an introduction, see arXiv:0702.365.
$^*$ However, some models with specific symmetries (for example SUSY) can be protected by some non-renormalization theorem, implying that the effective potential won't be renormalized. Hence, the generic terms one would expect are not present in the low-energy effective action.
$^{**}$ This is in fact true in most cases, but not always. In some specific model, it might happen that a symmetry is broken dynamically during the flow, for example if a singularity of the effective potential is generated along the RG trajectory (this implies doing some functional RG). In these rare cases, while the effective potential had a symmetry at high energy, it loses at low energy. For an example of this, in the context of the random field Ising model (where a (non-dynamical) supersymmetry is lost during the flow, see arXiv:1103.4812.
