We present numerical simulations aimed at exploring the effects of varying the sub-grid physics parameters on the evolution and the properties of the galaxy formed in a low-mass dark matter halo (similar to 7 x 10(10) h(-1) M(circle dot) at redshift z = 0). The simulations are run within a cosmological setting with a nominal resolution of 218 pc comoving and are stopped at z = 0.43. For simulations that cannot resolve individual molecular clouds, we propose the criterion that the threshold density for star formation, n(SF), should be chosen such that the column density of the star-forming cells equals the threshold value for molecule formation, N similar to 10(21) cm(-2), or similar to 8 M(circle dot) pc(-2). In all of our simulations, an extended old/intermediate-age stellar halo and a more compact younger stellar disk are formed, and in most cases, the halo's specific angular momentum is slightly larger than that of the galaxy, and sensitive to the SF/feedback parameters. We found that a non-negligible fraction of the halo stars are formed in situ in a spheroidal distribution. Changes in the sub-grid physics parameters affect significantly and in a complex way the evolution and properties of the galaxy: (1) lower threshold densities n(SF) produce larger stellar effective radii R(e), less peaked circular velocity curves V(c)(R), and greater amounts of low-density and hot gas in the disk mid-plane; (2) when stellar feedback is modeled by temporarily switching off radiative cooling in the star-forming regions, R(e) increases (by a factor of similar to 2 in our particular model), the circular velocity curve becomes flatter, and a complex multi-phase gaseous disk structure develops; (3) a more efficient local conversion of gas mass to stars, measured by a stellar particle mass distribution biased toward larger values, increases the strength of the feedback energy injection-driving outflows and inducing burstier SF histories; (4) if feedback is too strong, gas loss by galactic outflows-which are easier to produce in low-mass galaxies-interrupts SF, whose history becomes episodic; and (5) in all cases, the surface SF rate (SFR) versus the gas surface density correlation is steeper than the Kennicutt law but in agreement with observations in low surface brightness galaxies. The simulations exhibit two important shortcomings: the baryon fractions are higher, and the specific SFRs are much smaller, than observationally inferred values for redshifts similar to 0.4-1. These shortcomings pose a major challenge to the SF/feedback physics commonly applied in the.CDM-based galaxy formation simulations.