Szukać kogo? Szukać czego? (to search for whom/what?) = indeed it is the Genitive.
it is mój in Genitive for masculine nouns, and can be Accusative for masculine nouns ( buy not plate)
OK, this one indeed didn't have "dish" on the list of accepted answers, added now.
Is it common for singular nouns in genitive to have the same form as the plural of themselves when in nominative?
Well, yes, but it doesn't happen here...
It happens for most feminine nouns and for most neuter nouns. Genitive singular = Nominative plural = Accusative plural.
But plural of "talerz" is "talerze", so not here.
Why do we have two super critical alternatives to give the soft pronunciation z but here there is a z with no super=sign yet pronounced soft?
Okay, so I understand that the alternatives that you mean are "ż" and "rz" for the sound that could be rendered by English "zh"? Well, they used to be different a loooong time ago, now the difference is just an orthographical one.
Meanwhile ź is a completely different sound, 'softened z'.