Is the Standard Model consistent (UV complete)?
The Standard Model is consistent in perturbative expansions.
It is inconsistent non-perturbatively but all these inconsistencies only show up "qualitatively" at energies well above the Planck energy – where we know the non-gravitational Standard Model to be inapplicable, anyway.
The inconsistencies of the Standard Model involve the Landau poles – the $U(1)_Y$ hypercharge coupling $g$ diverges at a certain energy scale, due to the renormalization group running – and a similar problem with the quartic Higgs self-coupling (it would be a problem at low, below-Planckian energies if the Higgs mass were higher than those 200 GeV).
A perhaps more serious problem related to the latter is the instability of the Higgs vacuum. The minimum at $v=246\text{ GeV}$ in the Standard Model isn't really a global minimum for certain values of the Higgs mass $m_H$. The observed $m_H$ is lower than 130 GeV or so and for these values, the potential isn't stable. It has lower minima at vevs $h\gg 246\text{ GeV}$. To say the least, the Standard Model potential is metastable (lower global minima exist but one can get there only through a tunneling which occurs rarely) with a dangerously short lifetime. Whether the metastability (intermediate situation) is an inconsistency – when all cosmological considerations are taken into account – is debatable.
But if one is satisfied with predictions at energies lower than the Planck scale and up to the accuracy of relative corrections of order $E/m_{Pl}$, then the Standard Model may pretty much be put on lattice and the continuum limit will agree with the perturbative expansions and produce a consistent theory for all these phenomena (limited by energies and the error margin). In particular, all the UV divergences may be consistently subtracted and all the IR divergences only encode real physical phenomena and the situation when one has asked a wrong or sloppy question.
This (limited) consistency doesn't mean that one should believe that the Standard Model is actually the exactly right theory of Nature up to the Planck scale. There are many reasons to think it is not the case.
Indeed, the Standard Model is consistent in perturbative expansions, which acutally means that we do not really know if the Standard Model is consistent or not. So it is possible that the original Standard Model with 15 Weyl fermions per family is not consistent. In other words, there may not exist any well defined quantum model, whose low energy effective theory reproduce the original standard model. (Here a "well defined quantum model" has the following defining property: the dimension of the Hilbert space is finite for a space of finite volume, and the Hamiltonian operator acting on the Hilbert space contain only local interactions.)
The statements "The Standard Model is renormalizable and mathematically self-consistent" and "Standard Model is a well-defined theory, in the sense that everything is calculable" are not correct. The non-perturbative SU(2) instanton effects are not calculable at the moment. Perturbative expansions without instanton effects do not even conserve probability, and do not converge. Those fatal problems of the Standard Model are well known and have a name chiral fermion/gauge problem.
However, I have a recent work suggesting that a modified Standard Model with 16 Weyl fermions per family is consistent (ie UV complete). See arXiv:1305.1045 A lattice non-perturbative definition of an SO(10) chiral gauge theory and its induced standard model