Is it permissible for someone who is correcting the Imam in Tarawih prayer to step forward one pace from the first row to get closer to the Mushaf, then return to their place before bowing, leaving their spot vacant while they step forward, and without looking behind them when they return?

What this person does, opening the Mus'haf to follow the Imam in Taraweeh prayers, is permissible and there is no harm in it, especially if the Imam has a weak memory and needs someone to correct him. As for him stepping forward one pace to the Imam and then returning, it does not invalidate the prayer because it is a minor action done for the benefit of the prayer.