messages will always be processed immediately when they become visible in SQS, is that correct?

Immediately is a strong word, but for simplicity, I would say yes.

If a client has started the polling and a message got in the queue, sqs will send response back to client with this message. pollTimeoutSeconds will not affect response time. If a message is available, the call returns sooner than pollTimeoutSeconds

pollTimeoutSeconds is a same thing as WaitTimeSeconds parameter from ReceiveMessage https://docs.aws.amazon.com/AWSSimpleQueueService/latest/APIReference/API_ReceiveMessage.html