Learn Farsi: آکروبات جسارت و مهارت خود را روی میله به زیبایی نشان داد. - The acrobat displayed his boldness and skill on the bar beautifully.

This is exactly what your current lesson covers! In Farsi, you can frequently create adverbs of manner by combining the preposition be (meaning 'with' or 'to') with an abstract noun ending in -i. Here, zibâyi means 'beauty,' so be zibâyi literally means 'with beauty' or 'beautifully.' You can apply this productive pattern to many other words, such as be râhati (easily) or be sâdegi (simply).

Why does the direct object marker râ come after khod, and does khod refer to both 'boldness' and 'skill'?

Yes, khod (meaning 'self' or 'own') applies to the entire coordinated phrase jesârat va mahârat (boldness and skill). There is an invisible spoken ezafe linking the nouns to the pronoun, pronounced as mahârat-e khod. Because this entire phrase acts as the specific direct object that the acrobat displayed, the direct object marker râ must sit at the very end of the complete noun block.

Could I use the preposition bar instead of ruy-e for 'on the bar' in this sentence?

You could use bar, but ruy-e is the much more natural choice for modern, everyday Farsi when talking about physical objects. Bar tends to sound more formal or literary, whereas ruy-e (literally 'on the face of') is the standard way to express physical location on top of something like a bar.