Does špatny do double duty as bad/wrong or is that me losing the plot??
Yes, it does.
I'm confused why špatná is correct for a thing (street) and špatné is wrong when it is neutral.
'Ulice' is feminine. Grammatical genders often don't have anything in common with 'the real world'. Read the tips and notes. Or Wikipedia. Or check the recommended resources, like Naughton's Essential Grammar.
Question about the English language: When shall I use 'bad' and when is the adjective 'wrong' correct?
Wrong is not right.
Bad is not good.