Is this act of mine—hastening the audible prayer and reciting short surahs when performing the prayer, to avoid my wife hearing my recitation due to my shyness about my voice—considered showing off (riya') or associating partners with Allah (shirk)? And what is the expiation for that?

What you are doing by abandoning the Sunnah of prolonging the recitation [in prayer] for fear that your wife might hear you is incorrect and indicates riya' (showing off). Al-Fudayl ibn Iyad said: "Abandoning an action for the sake of people is riya', and performing an action for the sake of people is shirk (associating partners with Allah), and sincerity is that Allah protects you from both." Performing obligatory prayers at home without a valid excuse is not permissible, as congregational prayer is obligatory for men. The expiation for what you have done is sincere repentance and abandoning these transgressions.