Is one who prays alone behind the row, having entered the prayer while the Imam was bowing, and did not count the raka’ah due to doubt about catching it, and then did not pull another worshipper into the row due to hesitation, considered to be praying alone behind the row? And what is the ruling on his prayer?

You did well by not counting the rak'ah in which you doubted catching up with the imam. However, you erred by performing part of the prayer alone behind the imam and the ma'mum (follower). You should have stood to the right of the ma'mum or gently pulled him to stand with you. Your prayer is valid according to the majority of scholars, and even the Hanbalis, who invalidate the prayer of one who prays alone behind the row, qualify this by stating it applies to completing a full rak'ah, and you did not complete that rak'ah alone.