Learn Russian: Ребёнок хорошо развивается.

Not exactly, though it uses the same ending. In дом строится (the house is being built), the house is passively receiving the action from someone else. Here, ребёнок развивается is an intransitive action—the child is developing on their own. Adding -ся to a transitive verb like развивать (to develop something) makes it either passive or, as in this sentence, an action focused on the subject itself.

Can I use развивает without the -ся if I want to say the child "is developing"?

Only if you specify what the child is developing. Развивать (without -ся) is a transitive verb, meaning it needs a direct object. For example, ребёнок развивает талант (the child is developing a talent). Because this sentence means the child themselves is developing, you must use the reflexive form.

Why use the imperfective развивается here?

Because "is developing" is an ongoing, continuous process. The perfective equivalent, развиться, would mean "to have fully developed" as a completed result. Since the child is currently in the process of developing well, the imperfective aspect is required.

How is the -ется at the end of развивается pronounced?

The -тся part is pronounced like -ца (-tsa). As a quick refresher, whenever a verb ends in -тся or -ться, those letters blend together to make a sharp "ts" sound, so it sounds like "развиваеца".