Would a cat scan show the inflammation or swelling from primary amoebic meningoencephalitis? If there was signs would there then be a spinal tap?

Maybe. Ct is not the best test for early infectious processes in the brain. Mri with contrast is best, but much longer and not always available. If amebic encephalitis is suspected, a lumbar puncture (spinal tap) would be appropriate, even if the ct is normal.
Mri is better. Probably this is true. However and MRI i of the brain may be more sensitive. Either way a spinal tap would be confirmatory.