Can the random port be used while the Castle is under attack?

Yes and no. If you have a legion out of city attacking someone or you have sent a scout you have to wait until they return to use random tp. If not then yes you can use random tp to evade the attack. Random teleport is your best friend during SvS KE hit and runs.

Yes it can.

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No, you'll need to advance teleport first and then do random quick.