Yes the potential is put in "by hand". The SM Lagrangian is restricted by several principles that are believed to be fundamental, but those principles do not determine the Lagrangian uniquely. In particular, the dimensionless parameters are not determined but those principles, and nor is the particle content. But at the moment I don't know any experimental reason for believing that the potential is not "fundamental".
Yes there are some theoretical problems related to renormalisation, and that's not a surprise considering the weak mathematical grounding for the whole field of quantum gauge theories. As far as I know, the perturbation expansion that is used even in quite simple QFTs doesn't even converge.
The field theory itself respects the SU(2)xU(1) gauge symmetry. It is the solution of the field theory that breaks it down to a U(1) symmetry. The field theory is the "law of nature", which has some large symmetry group, and the "state of nature" is a solution to the field theory. Mathematically, there is no reason for the solution of a differential equation to have the full symmetry of the differential equation itself. Our world corresponds to a particular solution of the standard model field theory. It just happens to have a nonzero constant value of the Higgs scalar field, caused by the shape of the potential.