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)?
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.