Learn Farsi: این بیت یک معنی زیبا دارد.

Yes, she'r means poetry or a poem in general. A beyt specifically refers to a single couplet or two-line verse, which is the standard building block of classical Persian poetry.

How would I change this sentence to say "This verse has its own beautiful meaning" using the formal word khish from this lesson?

You would say in beyt ma'ni-ye ziba-ye khish ra darad. Notice that we had to add the direct object marker ra. This is because "its own beautiful meaning" is a specific, definite object, unlike the indefinite "a beautiful meaning" (yek ma'ni-ye ziba) in the original sentence.

Why is there a ye sound between ma'ni and ziba when spoken, even though it's not written?

This is the ezafe linking the noun (ma'ni) to its adjective (ziba). Because ma'ni ends in a vowel sound, the ezafe is pronounced as -ye instead of just -e. It is rarely written out, so you just have to know to pronounce it when you see a noun followed by an adjective.

Can the indefinite article yek be replaced by the suffix -i here?

Yes. To say "a beautiful meaning", you can use yek ma'ni-ye ziba as shown here, or attach the indefinite suffix -i to the end of the phrase to make . Both are perfectly natural, though using is very common in modern written Farsi.