Flippers and socks for small feet

If it's any help, I've done quite a lot of online research on traditional all-rubber fins, and some are available in a wide range of sizes, including children's sizes. One such fin is the Greek-made Majorca Sub Dolfin:

Swim Fins Junior for £8.68 at Newitts.com
Although not every size is available from Newitts, Majorca sub manufacture them from size 22 to size 48. They appear to be sold widely in Europe, including Scandinavia. They have soft foot pockets and flexible blades, unlike many plastic fins that come with sharp edges and blades that crack when dropped.

If you're after long-bladed fins, the size range is indeed much more modest. The smallest size I can find for the all-rubber Taiwan-made F702 long-bladed fin is 36-38:

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The problem with these fins is that the smallest sizes seem to be available only from an online retailer in Taiwan, where they are made.

I can't help with the socks, but I can say that in the pioneering days of the 1950s, youngsters often improvised by wearing ordinary socks or even soft plimsolls with fins that were too big when they went snorkelling. Good luck!