Learn Hardcore Russian: Я читаю газету утром.

Because газета is the direct object of читать, it takes the accusative case. Feminine nouns ending in -а replace -а with -у in the accusative singular, so газета → газету.

Why is утро in the form утром instead of using a preposition like “in”?

Russian often marks time of day by using the instrumental case without a preposition. The noun утро (“morning”) becomes утром in the instrumental to mean “in the morning.”

What tense and aspect is читаю, and how does that affect the meaning?

Читаю is present tense, imperfective aspect of the verb читать. Imperfective verbs focus on the process or habit of an action. Here it usually expresses a habitual action (“I read regularly in the morning”), though it could also mean “I am reading (right now).”

Could this sentence mean “I am reading the newspaper right now in the morning”?

Yes. The imperfective present can describe an action happening at this moment. Without extra context, it often implies a habit, but you could clarify by adding сейчас: Я сейчас читаю газету утром (“I am reading the newspaper right now in the morning”).