Learn Russian: Я бегаю каждый день.

Бегаю is a multidirectional verb (from бегать). In your current lesson, you are learning how to use verbs like ходить for repeated habits. Бегать works the exact same way! Because the sentence says каждый день (every day), we need the habitual, multidirectional verb, not the one-way verb бежать (which means running in one specific direction right now).

Is the phrase каждый день in the nominative case here?

It looks exactly like the dictionary form (nominative), but it is actually in the accusative case. In Russian, we use the accusative case for time expressions that answer the question "how often?". Because день (day) is an inanimate masculine noun, its accusative form doesn't change from its nominative form.

If I wanted to say "I run every morning" instead, would the word каждый stay the same?

No, каждый (every) acts like an adjective and has to match the gender of the noun it describes. День is masculine, so we use the masculine form каждый. But "morning" (утро) is a neuter noun, so "every morning" would become каждое утро.

Does бегать have tricky spelling changes like some other verbs of motion?