Hi Steve
If that's your Dad in the pictures he was definately serving in the Royal Navy!
The 'butterfly-like' badge you describe on his right arm is his 'trade' badge, denoting which branch of the Navy he served in. He was in communications, as this is the Telegraphists badge. It consists of a pair of wings crossed by a lightning flash (see image below). All trade badges, regardless of what branch a seaman served in, were worn on the right arm.
What you describe as a decorative badge on his cap is actually the bow which tied the cap tally (ribbon) onto the cap. Officially the bow should have been worn above the left ear but during the war it became fashionable for ratings to wear the bow above the left eye! Although in contravention of regulations this was tolerated at the time.
I hope this makes things a bit clearer for you.
Pete
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