The English name is still "the Czech Republic", and most Czechs (myself included) prefer to use it instead of "Czechia".
"Já bydlím v Česku" is the correct spelling.
"z Česka" is the correct form, "od Česku" doesn't make any sense.
That being said, "Česká republika" is actually used quite often. "Česko" is more colloquial, and wouldn't probably be used in certain situations (like at school, for example).
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