Is it obligatory to renew the marriage contract after doubting apostasy due to: leaving a newspaper containing the name of Allah the Almighty (without being certain of the presence of Allah's name), or leaving papers containing the name of Allah in a public place? And is repeating the Shahadatayn with the intention of renewing faith and repenting for any unintentional disbelief sufficient?

Placing kitchen utensils on papers bearing the name of Allah is not an act of disbelief (kufr), especially since you are not certain if they contain the name of Allah. Your request to your wife to remove them, your great concern over the act, and your fear of disbelief all distance you from disbelief. What you have described does not go beyond the realm of obsessive whisperings (waswasah), and you should disregard them. Your wife is still lawfully bound to you. You are far from apostasy (riddah), and these are merely whisperings. In the future, you can bury these papers if you are unable to burn them.