Learn Russian: Женщина позёвывает и почитывает старый журнал. - The woman yawns and casually reads an old magazine.

It adds the idea of doing an action occasionally, casually, or with low intensity. Instead of regular yawning (зевать), позёвывает means she is yawning lightly or from time to time. Similarly, почитывает shows she is just skimming or reading casually, rather than being deeply engrossed in the text.

Are позёвывает and почитывает perfective or imperfective? Doesn't the prefix по- usually make a verb perfective?

They are imperfective. You are right that по- usually creates perfective verbs (like почитать). However, the addition of the -ыва- or -ива- suffix acts as an imperfectivizer, bringing the verb back to the imperfective aspect. This is why these verbs can be used in the present tense to describe what the woman is currently doing.

Why does one verb use -ива- (почитывает) while the other uses -ыва- (позёвывает)?

The choice depends on the final consonant of the verb root before the suffix. After soft consonants and hushers, Russian uses -ива- (as seen in почитывает). After hard consonants, it uses -ыва- (as seen after the hard consonant in позёвывает).

Why is старый журнал in the nominative form if it is the object being read?