Learn Russian: Она твёрдо отстаивает своё мнение.

In Russian, when the possessor is the same as the subject of the sentence (here, Она), you use the reflexive possessive pronoun свой (in this case, the neuter своё to match мнение). If you used её, it would strictly mean she is defending another woman's opinion.

How does the structure of the verb отстаивает help explain its meaning?

The imperfective verb отстаивать comes from the root стоять (to stand). The prefix от- often implies fending something off. Together, they create the idea of "standing your ground" or defending a position, which is perfect for standing up for an opinion or your rights.

Why doesn't the ending of своё мнение change, given that it is the direct object?

It actually is in the accusative case here! Because мнение is a neuter, inanimate noun, its accusative form is completely identical to its nominative (dictionary) form.

The current lesson covers particles like всё-таки and всё же (nevertheless / still). How would you add one to say "She firmly stands up for her opinion anyway"?

You could say Она всё-таки твёрдо отстаивает своё мнение or Она всё же.... Placing the particle right after the subject emphasizes that despite some unstated opposition or previous context, she continues to defend her stance.