Actually I myself would find "skarpetki" to be the basic form and "skarpety" to be some thick, big, woolen socks for when it's really cold.
But technically, "skarpetki" is the diminutive, so probably that caused "skarpety" to be used here. Or maybe it just depends on a person, which word he or she prefers.
I thought it was "these socks are old" and not "those socks are old" but is there that much of a difference between the two?
The problem is that English and Polish look at determiners quite differently. Polish uses "te/te/tamte" (and its forms) while English uses "these/those/those". That means that "those" is a lot more common than Polish "tamte". Therefore if the Polish sentence did use "tamte", it really is "those ones over there" and only "those" can be accepted.
The adjective has to match the noun it describes. "Skarpety" are 'not masculine-personal' plural. So the right form of the adjective is the same as for neuter singular: "stare".
"stary" is masculine singular.
It's hard to learn Polish with a Russian background. "СтарЫЕ" ) It has both "y" and "e".
Yeah, 'the' should be acceptable for 'te', among with 'these' (main answer) and 'those'; but for 'tamte' only 'those' should work. Removed now.