Learn Farsi: دوچرخه‌ی پسر جدید است.

Because the word docharxe (bicycle) ends in a vowel sound. In Farsi, when a noun ends in a vowel, the Ezafe particle becomes -ye to make it easier to pronounce. If the word ended in a consonant, you would just add an -e sound to connect it to the next word.

Why is the order docharxe-ye pesar and not pesar-e docharxe?

In English, we often put the owner first using an apostrophe-s ('the boy's bicycle'). In Farsi, possession works like the English phrase 'the bicycle of the boy'. The thing being owned (docharxe) comes first, followed by the Ezafe connector (-ye), and finally the owner (pesar).

The English translation says 'The boy's bicycle', but where is the word for 'the' in Farsi?

Farsi does not have a direct word for the definite article 'the'. A word like pesar can mean 'a boy' or 'the boy' depending on the context. Because we are talking about a specific owner's bicycle here, it naturally translates to 'the boy' in English.

Why is the verb ast (is) at the very end?

In Farsi, the verb almost always goes at the very end of the sentence. The literal word order of this sentence is 'Bicycle-of boy new is.'