Learn Arabic (Modern Standard): يمشي مشية واثقة.

It uses the Noun of Manner pattern فِعْلَة (fi'lah). We take the root of the verb يمشي (which is م-ش-ي) and apply this pattern. Because the first letter takes a kasra (i), it becomes مِشْيَة (mishyah). This specific pattern is used in Arabic to describe the physical state or manner in which an action is performed.

Why does the sentence repeat the root letters by placing مشية right after the verb يمشي?

This is a standard Arabic structure called the absolute object or cognate accusative (مفعول مطلق). Where English typically uses an adverb ("He walks confidently"), Arabic frequently pairs a verb with a noun derived from its own root to describe the action. Literally, the structure translates to "He walks a confident walking."

What case endings would مشية واثقة take in formal, fully voweled Arabic?

They would both take the accusative ending with tanwin (the "-an" sound), appearing as مِشْيَةً واثِقَةً. The word مشية is accusative because it is the absolute object of the verb. The word واثقة (confident) is an adjective, so it must perfectly match the noun it modifies in gender (feminine), number (singular), and case (accusative).