Why is my Breit-Wigner resonance shape so asymmetric, with a long tail down to low masses? In some cases I even see a secondary peak at the lower mass cutoff.

The Breit-Wigner shape is convoluted with the incoming flux provided by the parton density functions. For hadron beams these are strongly peaked at small x values, i.e. they favour the production of low-mass systems. This skews the Breit-Wigner and can even give rise to a secondary peak (see further the PYTHIA manual section 7.3.) However, you should not always take the tails too seriously; often there are other Feynman graphs contributing to the same final state, which can be neglected in the Breit-Wigner peak region but not in the wings of the distribution, so that a more careful description is required there.