Learn Russian: Я купил шкатулку, чтобы подарить ей кольцо. - I bought a jewelry box to give her a ring.

We use чтобы (in order to) plus an infinitive verb when the person doing the action in both parts of the sentence is the same. Since the same person is buying the box and giving the ring, подарить stays in the infinitive. If you bought it so someone else could give the ring, you would use чтобы with the past tense.

Why is the pronoun ей (her) used here instead of the Accusative form её?

The verb подарить (to give as a gift) takes two objects: the thing being given, and the recipient. The recipient always takes the Dative case because the action is directed to them. ей is the Dative form of она (she).

Since кольцо (ring) is the thing being given, shouldn't its ending change to show it is the direct object?

It actually is in the Accusative case here, but its ending does not change! кольцо is a neuter noun, and in Russian, neuter nouns look exactly the same in the Accusative case as they do in their dictionary (Nominative) form.

Why does the word for jewelry box (шкатулка) change to шкатулку?

It is the direct object of the verb купил (bought), so it takes the Accusative case. For feminine nouns ending in -а, the ending changes to -у in the Accusative.