What is the ruling on marrying a foreign woman via the Internet, what is the ruling on the relationship that resulted from a marriage contract without witnesses, what is the ruling on the child resulting from this relationship, is it permissible to name him Muhammad, and what are the conditions for sincere repentance?

It is clear that what happened was an abomination due to being lax with a foreign woman. You have done well by repenting, and the conditions for repentance are found in Fatwa no. 32521.

You must be cautious in the future and increase in righteous deeds. This marriage contract is invalid due to the absence of the guardian's permission and the presence of witnesses, so it must be annulled.

The child is attributed to you due to the doubt, as long as you believed the intercourse was permissible with the aforementioned contract. Your vow to name him Muhammad is a permissible vow, so it is not obligatory to fulfill it. If you do not fulfill it, then as a precaution, offer an expiation for breaking an oath.

The child is a Muslim and follows your religion. There is no problem in naming him Muhammad.