Import of dogs and cats air by manifested cargo only and travel in the belly, think it would be the same for other furry live animals.
Know about dogs and cats also need a import permit.
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About apartments on the economy (not EK), many say no pets to start with but will allow cats without issue, just note it on the contract.
The company we rent from has no pets in the standard contract but allows cats only in our building and in the next building over small dogs and cats are allowed.