Does a transaction always get verified/processed by ALL full nodes in the network before being added into the block by some mining node(s)? If not, then why is it often claimed that the blockchain scaling issue is due to the fact that all tx has to be verified by all full nodes?   

As long as a mining node found the correct nonce, the tx will be added to the block and the blockchain advances. It does not matter if the tx is not yet verified by some full nodes in the network.   

(Yes, it will most probably get verified by all full nodes after it got into the block and the block propagates throughout the network. But this is not relevant as the blockchain already advanced (assuming there is no fork).)
Does a transaction always get verified and processed by ALL full nodes in the network before it is added into the block by some mining node(s)?


Technically no, but only because of the way you've phrased the question. In a decentrilized network, there is no way to know what ALL nodes have done.

However, EACH particular full-node (including miners) will verify every transaction in a candidate block before updating their own blockchain state to the new head. Since no one will include the invalid block, it doesn't accomplish anything to produce one.