Do the rewards for acts of obedience and worship go to someone other than their doer because of involuntary whispers and obsessions? And what should one do to repel these whispers?

You must completely ignore these whisperings (waswas) and pay no attention to them, as this is the best remedy for them, in addition to ruqyah (Islamic exorcism) and the Prophetic ذكر (dhikr - remembrance of God). Your good deeds are for you, and you will not be held accountable for the sins of others, for Allah Almighty says: "Whoever is guided is only guided for [the benefit of] his soul. And whoever errs only errs against it. And no bearer of burdens will bear the burden of another." What you find within yourself or utter due to these whisperings will not diminish your reward.