Zvlastni = strange/weird, odd/unusual. Can it also mean unfamiliar? In English we can say "I visited a strange city last week" to mean odd, weird OR unfamiliar to me, unknown to me. Can zvlastni have this meaning?
No. Strange in that meaning of "unknown" is "cizí". CIZÍ can also mean foreign.
Why is it zvláštní rather than zvláštné?
I thought Neuter nouns take -é endings.
You are correct. That rule goes for "hard" adjectives. There are two kinds hard and soft. Soft end in í nearly exclusively, if not exclusively, and these do not change gender. So you can have
zvláštní žena. zvláštní muž, zvláštní dítě...
"zvlastni" can also be interpreted as "important" or "special"?
"Special" in some cases, like "special effects" = "zvláštní efekty".
"Important" not really.
Is there another word for a smaller 'city'? Most places I see referred to as a 'město' are actually what I would call a town. In the Czech Republic, I would only really call Prague a city.
There is, but firstly, the capitals of the Czech administrative regions are definitely cities, secondly, even around 10000 inh. it is still město. A small town will be městečko.