Learn Russian: Она осторожно приподнимает край ковра. - She carefully lifts the edge of the carpet a little.

The prefix при- adds an attenuative meaning here, which means doing an action only slightly or incompletely. Instead of lifting the carpet fully, she is just lifting it a little bit (приподнимает).

Is приподнимает perfective or imperfective, and what is its verb pair?

It is imperfective, as it describes a present action. Its perfective counterpart is приподнять. It builds on the standard pair поднимать (imperfective) and поднять (perfective) by simply adding the при- prefix to both forms.

Край is the direct object being lifted. Why hasn't its ending changed for the accusative case?

Because край (edge) is a masculine, inanimate noun. In Russian, masculine inanimate nouns share identical forms for the nominative and accusative cases. If it were an animate noun, the accusative ending would change to match the genitive.

Why does the base word for carpet (ковёр) change to ковра here, dropping the vowel?

Ковёр is in the genitive case here because it translates to of the carpet. It contains a fleeting vowel (the ё). When grammatical endings are added to this word, that vowel drops out. You see this same pattern in familiar words like день (дня) or отец (отца).