What is the ruling on children born from an illegitimate relationship (cohabitation)? Are they considered children of "zina" (fornication/adultery)? And what is the ruling on their guardianship for someone who intends to marry their mother and wants to proselytize in a society that includes Muslims and non-Muslims?

If the two children mentioned are children born out of wedlock (zina), then they are attributed to their mother. Their guardianship and showing kindness to them is a great act of nearness to Allah. The intention to invite people to Allah and to make them worship their Lord is one of the best acts of obedience, and sometimes it can be more impactful than actions themselves. As for traveling to lands of disbelief, its dangers are great, and it is not permissible to reside there except out of necessity or severe need, such as for the purpose of da'wah (inviting to Islam). However, one should be cautious and ensure protection from temptation (fitna). As for marrying the woman mentioned, do not rush into it until you ask about her from trustworthy people and confirm her repentance and good conduct. If the marriage does not take place, then sever all ties with her.