Is eating meat slaughtered for other than God, about which there is doubt – after the fact – subject to the rule "doubt does not nullify certainty" or the Hadith "whoever guards against doubtful matters has indeed protected his religion and his honor"? And is expiation obligatory if it was unlawful or if there was doubt concerning it?

If the doubt about your Muslim neighbor's slaughtered animal has no realistic justification, then the basic rule is that it is permissible (halal), because he is a Muslim who slaughters according to the Shariah method. However, if the doubt is justified, such as if you know that he slaughters for other than Allah, then two principles conflict: that the basic rule for a Muslim is to slaughter for Allah, and that the basic rule for slaughtered animals is prohibition (haram) when there is doubt, which necessitates their prohibition. In case two principles conflict, the more cautious approach is taken, which is to refrain from eating from the slaughtered animal of someone known to slaughter for other than Allah, while advising him and reminding him of Allah. You are not obligated to offer expiation for past consumption, but be cautious in the future.