Learn Spanish (Spain): ¿A quién le regalarías esta joya?

It is the conditional tense for tú (you). To form regular verbs in the conditional, you take the full infinitive regalar and add the conditional ending for tú, which is -ías. Unlike English, which adds the separate word 'would', Spanish attaches the ending directly to the infinitive.

Why is there a le if the sentence already asks A quién (to whom)?

This is an example of indirect object duplication, a very common feature in Spanish. When the indirect object (the person receiving the action, here A quién) appears before the verb, you must also include the matching indirect object pronoun (le).

Why does the question start with A instead of putting the preposition at the end like the English 'to'?

In Spanish, a sentence or question can never end with a preposition. The preposition a (to) must go directly in front of the question word quién (who), making the literal structure 'To whom would you give this jewel?'.

Can I use the verb dar (to give) instead of regalar in this sentence?

You could, but regalar is much more precise. While dar is the general verb for 'to give', regalar specifically means 'to give as a gift'. Since una joya (a jewel) is a typical present, using regalarías sounds much more natural here.