Yup, Weimar eagle. Nice and clear too. The letter after the serial number is the block, they would go from 1 to 9999, then start over 1 to 9999a, then 9999b, and on up. Yours was "g" block, and assuming it is a S/42, they went to Z block in 1936, so yours was probably made in March or April.
You don't have to take a photo, but can you tell me any numbers on the Eagle/number inspection stamps, like the one on the floor plate?
It is definitely not a bad idea to have the headspace checked, but if it was me, I would take it out and shoot it.
The history would be kind of cool to know, did you have any relatives that fought in the war? If your Dad was born in 42, his father and maybe uncles could be the right age. Or, he could have picked it up from a barrel in a hardware store in the 50s, then had it converted to a hunting rifle.
Lots of military rifles were modified back then to make them more suitable as hunting rifles in the years after WW2, simply because a new commercial Remington or Winchester was pretty expensive, possibly a few weeks wages, where you could pick a surplus rifle for ~$10.
And get to work on that rust.....
Really sorry about your Dad, that is way too young. You only getting the scraps of your Dad's guns is a load of
Scheiß
Oh, this is from me to your step mom and her son....