Is saying, "Come, let us pray to see who prays faster," considered a sin, mockery of prayer, or apostasy, given that the intention is for a valid prayer? And what is the manner of repentance if it is a sin?

You are afflicted with obsessive whispers (waswasah). The cure for it is to completely disregard it and to increase in remembrance of Allah (dhikr), seeking forgiveness (istighfar), and seeking refuge (ta'awwudh). Whoever succumbs to it will find it increasing until it drives them to the brink of madness. Furthermore, there is no harm in performing prayer quickly as long as tranquility (طمأنينة) is present. However, if it compromises a pillar of the prayer, then the prayer is invalidated.