Learn Turkish: Çip zararsız olmasına rağmen kuşlar halkayı beğenmedi. - Although the chip is harmless, the birds did not like the ring.

It uses the exact same verbal noun logic! The verb olmak (to be) becomes the verbal noun olma (being), and gets the 3rd-person possessive -sı (its being). The difference is the postposition. While için doesn't take a case ending, rağmen (despite/although) always requires the dative case (-e/-a). So we add the buffer -n- and the dative -a to get olmasına rağmen (literally 'despite its being').

If it translates to 'despite its being', shouldn't the word 'çip' have a genitive ending like 'çipin'?

You are completely right! Strictly speaking, it is a genitive-possessive pair: çipin ... olmasına. However, in everyday Turkish, the genitive ending on the subject of these clauses is frequently dropped, especially for inanimate objects or general concepts. The possessive ending -sı on olmasına is enough to link the action back to the chip.

Why is the word for ring, 'halka', written as 'halkayı' here?

Because the birds didn't just dislike rings in general; they disliked the specific ring placed on them. The verb beğenmek (to like) takes a direct object. To point to a specific object, we add the accusative case ending -ı. The -y- is a buffer letter to separate the two vowels.

How is the word 'zararsız' (harmless) built?