Yes a big group of Greek verbs are conjugated in passive but have an active voice. Some call them medio-passive verbs. The maybe most common verb to be/ είμαι belong to this group. To become is a better example because it has the ordinary endings in all persons: γίν-ομαι, γίν-εσαι, γίν-εται, γιν-όμαστε, γίν-εστε, γίν-ονται.
Yes. Grammarians (particularly those who study both ancient and modern Greek) say Greek has "active" vs "medio-passive". "Medio" refers to the "middle voice" where one does something that primarily affects oneself. Hence a lot of English intransitive verbs line up with Greek middle voice (or "passive conjugation with active meaning") like come <-> έρχομαι, become <-> γίνομαι and so on. You can find a lot more guidance by searching "Greek middle voice".