I'm not sure there is a definite answer, since nobody seems to agree as to what QFT really is (at least so says Nate Seiberg in his 2015 Breakthrough Prize talk), but the problems you mention are well known and accounted for.
Quantum Mechanics gets loads of issues with infinite degrees of freedom. So in reality what we actually do is to regularize the theory, both infrared (e.g. putting in finite volume) and ultraviolet (e.g. putting in a lattice). When properly regularized QFT has finite degrees of freedom, and therefore all Haag's objections go away. You have a perfectly defined interaction picture and a mapping between free and interacting ground states, so everything goes smoothly.
This has the drawback of breaking Lorentz Symmetry, but the hope is that once we renormalize, by taking appropriate limits, we are left with sensible answers and recover Poincaré invariance.
Since the couplings in QFT are very singular and ill-defined we should have done the regularization in the first place just to know what the hamiltonian is, so this is a key part of QFT.
The problem with Haag's approach, and other axiomatic ones, is that they work directly with already renormalized fields, and therefore run on multiple problems. This is why more recent textbooks, starting with Weinberg (I think), emphasize that regularization and the renormalization group are key concepts one cannot do QFT without, even in the absence of interactions.
This post imported from StackExchange Physics at 2017-10-11 16:30 (UTC), posted by SE-user cesaruliana