Learn Spanish (Latin America): Movieron el andamio precario y empezó a tambalearse. - They moved the precarious scaffold and it started to wobble.

No, in Spanish, to say you "start doing something," you must use empezar a followed by an infinitive. Therefore, it has to be empezó a tambalearse. If you said empezó tambaleándose, it would mean it started its sequence of actions by wobbling.

Why does tambalearse have the pronoun se attached to it?

The verb tambalearse (to wobble or stagger) is pronominal here, meaning it requires the pronoun to show that the subject itself is experiencing the movement. Without the se, the verb tambalear means to push or make something else wobble.

Could I move the se from the end of tambalearse to another part of the sentence?

Yes! Because empezó a tambalearse functions as a single verbal unit (a conjugated verb plus an infinitive), you can place the pronoun before the conjugated verb instead. Saying y se empezó a tambalear is perfectly natural and means the exact same thing.

Why is the preterite tense used for both movieron and empezó?

The preterite is used to narrate a chain of consecutive, completed events. One action finished (they moved the scaffold), which immediately triggered the next specific action (it started to wobble). Using the imperfect tense here would incorrectly describe continuous background actions rather than a timeline of events.