Are the marriages of the disbelievers regarded as valid for them and are the children to be attributed to them?

Firstly: 

Marriages that are regarded by the disbelievers as valid are
approved by Islam and deemed to be valid, and the same consequences apply to
them as to marriages that are sound according to sharee‘ah, such as
legitimacy of children and inheritance between the spouses, and other
rulings connected to valid marriage. 

At the time of the Prophet (blessings and peace of Allah be
upon him) many people became Muslim and he did not ask about the details or
conditions of their marriages; rather he approved of them, which indicated
that they were valid.  

Ibn Qudaamah (may Allah have mercy on him) said: The
marriages of the disbelievers are valid and are approved of if they become
Muslim or if they refer to us for judgement, if the woman is one whom it is
permissible to marry at the present moment, and the details and manner of
the marriage contract are not to be examined and the marriage is not to be
examined to see whether it conformed to the conditions for marriage among
the Muslims, such as the presence of the wali (woman’s guardian) and
witnesses, and the format of the proposal and acceptance, and so on. There
is no difference of opinion concerning this among the Muslims. Ibn ‘Abd
al-Barr said: The scholars are unanimously agreed that if two spouses become
Muslim together at the same time, they remain married according to their
original marriage, so long as they are not too closely related by blood or
through breastfeeding. People became Muslim at the time of the Messenger of
Allah (blessings and peace of Allah be upon him) and their wives also became
Muslim, and their marriages were approved of; the Messenger of Allah
(blessings and peace of Allah be upon him) did not ask them about the
conditions of their marriage or how it was done. This is something that is
known through mutawaatir reports and is something that is well known to all
Muslims, so it is certain. But the marriage may be checked in some ways. If
the woman is one whom it is permissible for him to marry in the first place,
the marriage is approved, but if she is one whom it is not permissible for
him to marry in the first place, such as a mahram (close relative) through
blood or otherwise, or because she is in ‘iddah, an apostate, an idol
worshipper, a Magian, or one whom he has thrice divorced, then it is not to
be approved. If they got married during the ‘iddah and they became Muslim
after the ‘iddah ended, then it is to be approved, because it is permissible
for him to marry her in principle. 

End quote from al-Mughni, 7/115 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:
Marriages of the disbelievers come under the same rulings as marriages of
Muslims with regard to consequences such as zihaar, li‘aan, talaaq,
attribution of lineage, and so on. 

Secondly: 

If the marriage is valid according to Islamic sharee‘ah, then
it is valid. If it is invalid according to Islamic sharee‘ah, then it may be
approved subject to two conditions:

(i)               
that they regarded it as valid
according to their own laws

(ii)             
that they did not refer to us
for judgement. 

If they do not regard it as valid, then they should be
separated. If they refer to us for judgement, we should examine the case. If
that is before the marriage contract has been done, then we should do it
according to our laws. If it is after the marriage contract has been done,
then we should find out whether the woman was permissible (for him to
marry), in which case we should approve the marriage, but if she was not
permissible then we should separate them. The evidence for these things is
the fact that disbelievers became Muslim at the time of the Prophet
(blessings and peace of Allah be upon him) and he left those who were
married on the basis of their marriages during the Jaahiliyyah and did not
discuss the matter. This indicates that (such marriages) are to be left as
they are.

End quote from ash-Sharh al-Mumti‘, 12/239 

Based on that, the children who are born to non-Muslims, if
they were born within a marriage that they believe is a valid marriage, are
indeed their children and attributing them to them is valid. 

But if they did not believe that this was a proper marriage;
rather it was a kind of friendship and love without marriage, then these are
illegitimate children, not legitimate. 

But if the zaani attributes them to himself and regards them
as his children, then they are his children and are to be attributed to
him. 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked
about the problem of zina (fornication or adultery) which few of our
brothers and sisters were free of before they became Muslim; and many
children resulted from these relationships. With regard to the children who
are born as a result of zina (fornication), is it obligatory for their
fathers who have become Muslim and who were not married to the mothers to
spend on their maintenance? 

He replied: With regard to these people who had intercourse
when they were still disbelievers, if they believed that this intercourse
occurred in the context of a marriage contract that they thought was valid,
even though it is invalid according to sharee‘ah, the marriage contract is
valid and the children belong to that man. For example, when he was still a
disbeliever, a man proposed to a woman that he should be her husband and she
agreed, and they thought that this was a marriage contract, then the man and
woman became Muslim. We say: Your marriage is still valid and there is no
need to repeat the marriage contract, and any children that you had are your
children, unless the wife is not permissible for the husband in Islam. An
example of that is if he was a Magian and married his sister, as the Magians
regard it as permissible to marry mahrams (close relatives). If he married
his sister when he was a disbeliever, then he became Muslim and she became
Muslim, they must be separated because the woman is not permissible for the
man. With regard to the group of people you mention we say: If they believe
that what they did of having relationships with these women was a marriage,
then it is not zina and the children belong to them. But if you believed
that it was zina and you attributed these children to yourself when you were
still a disbeliever — i.e., the zaani said: These are my children — then
they are his children, so long as there is no one to dispute with him on
this issue. But if they did not attribute these children to themselves, then
they are not their children. 

With regard to maintenance, if we determine that they are
their children, then they are obliged to spend on their maintenance; if we
have not determined that, then they are not obliged to spend on their
maintenance. 

End quote from Fataawa Noor ‘ala ad-Darb.