Which is better and more greatly rewarded: praying Asr and then remaining in the mosque until Maghrib, or returning home before Maghrib, renewing ablution, and then going out to the mosque to renew the reward for steps?

It is better for a person to remain in the mosque, remembering Allah, for the reward of staying is greater than the reward of walking to it. The angels pray for him and supplicate for his forgiveness and mercy as long as he remains in his prayer place, and he is considered to be in prayer as long as he is waiting for it. This is preferable to him leaving to perform ablution and then returning. Furthermore, his remaining protects him from idle talk, for "a prayer followed by another prayer, with no idle talk in between, is recorded in Illiyyun."