What is the expiation for falling into doubts about the existence of God, and for backbiting and slandering people who cannot be seen now, after the age of twenty and with the resolve not to return to these sins?

What you mentioned about doubts concerning the existence of Allah the Almighty are mere whisperings (waswas) that cause no harm. You must strive to combat them and try to rid yourself of them, and read accessible books on creed (aqidah).

As for backbiting (gheebah) and slander (buhtan), you must repent and feel remorse for your sins, and increase your seeking of forgiveness for those you backbit about, and supplicate for their well-being.