"Turn on point" for fusion?
I don't have detailed model calculations to hand and I'm not sure they have been done for the small range of metallicity you mention. However, in handwaving terms.

(1) Mass and composition are the most important variables. Helium abundance, in addition to metallicity, could play a role. Rotation could be a very second order effect. Very rapid rotation would lower core temperatures at the same mass.

(2) This is the general assumption if a star plus brown dwarf companion formed in a binary or even if they formed in the same cluster.

(3) Higher metallicity leads to larger opacity in the atmosphere and a larger brown dwarf. This is a small effect in fully convective objects like these, except perhaps when they are very young and have superadiabatic surface layer. However, as this is when D burning takes place, possibly there is a small effect - higher metallicity leads to a larger radius for the same mass and a lower core temperature. So the mass threshold would be slightly higher.

(4) The duration of D burning is mass dependent. The graph below, from Burrows et al. (1997), shows luminosity vs time for brown dwarfs, planets and stars. The D burning phase is the early plateau in luminosity, lasting a few million years for things which are almost stars to maybe one or two hundred million years at 13 Jupiter masses. When D fusion ceases, then cooling recommences and the luminosity will continue to fade (forever), unless the object is massive enough to ignite hydrogen (i.e. a star: the difference between the green lines and blue lines on the plot).