In a nucleophilic substitution reaction, does the other atom to which the nucleophile is bonded matter?
When undergoing an SN rxn, does the other atom to which the
  nucleophile is ionically bonded matter?


It can matter.  If the counterion shifts the covalent-ionic nature of the bond connecting it to the nucleophile, then the nucleophilicity of the nucleophile can change (increase or decrease).  However in your example, the counterions don't make a significant difference.


  Which is more valid, and why?


On paper, they are both valid but

$\ce{H2S}$ is very toxic and corrosive, so the first method using sodium hydrosulfide would likely be preferred.  However, in industry, where raw material cost is very important, both of these methods are successfully used.

However both methods suffer the same drawback - once some of the desired thiol is prepared, it can react further with another molecule of alkyl bromide to produce a thioether; usually a mixture of thiol and thioether is obtained.  

In the lab, neither method is used

In the chem lab, the preferred method to prepare simple thiols is through the use of thiourea.  Note the resonance structures for thiourea (just disregard the lower half of the diagram) that place negative charge on the sulfur, making it a good nucleophile.



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In the first step the thiourea reacts with the alkyl halide via an $\mathrm{S_{N}2}$ reaction to yield an isothiouronium salt, which is subsequently hydrolyzed by sodium hydroxide to generate the thiol. This synthesis can be carried as a one pot, two step reaction.



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