First of all, children are not by-product of a marriage, they are blessings from God, just like we are not by-products of God's creation :)
Answering your question regarding 1 Cor 7:14
Now to answer your question, let's look at the context of the verse.
Paul has discussed marriage related topics with 2 categories of people:
- Believers who were not married at that time (1 Cor 7:8-9), quite self-explanatory:
8 But I say to the unmarried and to the widows: It is good for them if they remain even as I am; 9 but if they cannot exercise self-control, let them marry. For it is better to marry than to burn with passion. (NKJV)
- Believers who were already married (1 Cor 7:10-11):
10 Now to the married I command, yet not I but the Lord: A wife is not to depart from her husband. 11 But even if she does depart, let her remain unmarried or be reconciled to her husband. And a husband is not to divorce his wife. (NKJV)
Note the phrase I command, yet not I but the Lord, which clearly states that this is a command from the Lord (Therefore what God has joined together, let no one separate.
)
Now the Corinthians would naturally have questions like: what if my spouse is an unbeliever? And that's what Paul tried to address in the later verses:
12 But to the rest I, not the Lord, say: If any brother has a wife who does not believe, and she is willing to live with him, let him not divorce her. 13 And a woman who has a husband who does not believe, if he is willing to live with her, let her not divorce him. (NKJV)
Note the phrase I, not the Lord, say, which states that this is from Paul and it's not a commandment directly from the Lord. Now let me make it absolutely clear, I do NOT agree that any believer should divorce the unbeliever spouse, because it violates what the Lord has commanded.
In this letter's context, Paul is trying to explain as clearly as possible why believers should not divorce their unbeliever spouses who are willing to stay in the marriage. He could have simply said something like: it's what the Lord has commanded, just do it, whether you like it or not is irrelevant.
But instead he's sharing a beautiful message:
14 For the unbelieving husband is sanctified by the wife, and the unbelieving wife is sanctified by the husband; otherwise your children would be unclean, but now they are holy. (NKJV)
The Greek word ἡγίασται
(sanctified) is used with perfect tense, which means the unbeliever spouses are already declared clean because they stay in the marriage with a believer spouse.
Does this means ultimate salvation for the unbeliever spouse? The scripture clearly teaches that to be saved you need to believe in Jesus, repent, get baptised in the name of the Father, the Son and the Holy Spirit, and not abandon your faith in Jesus Christ. Salvation is between God and each individual, you can NOT be saved as an unbeliever just because you have a believer family member.
Also, it feels strange if Paul talked about marriage in the previous verses and following verses, and suddenly talked about the ultimate salvation of the unbeliever spouses in the middle.
My personal opinion here:
So what does Paul mean here? Given the whole chapter is mostly about single life / married life, so in this context it is very likely that, instead of discussing the salvation of the unbeliever spouse, it's much more an encouraging message to the believer spouse that the marriage is a valid marriage in the eyes of God and should be respected. It is as valid as a marriage between two believers, and the children too are acknowledged by God (again, does not necessarily mean salvation).
Naturally the Corinthians would then wonder: what if the unbeliever spouse chose to leave me? Paul basically said, don't worry about it, leave it with God and have peace (note Paul's NOT saying the unbelievers won't be saved)
15 But if the unbeliever departs, let him depart; a brother or a sister is not under bondage in such cases. But God has called us to peace. (NKJV)
Why is that? Because God has an individualised plan for everyone, you don't have the ability or duty to save the leaving unbeliever spouse (and I personally add here - "or your children"), salvation is the work of God with each individual.
Here's the explanation (and Paul off-tracked a bit to the whole church before coming back to the marriage topics and talk about virgins):
16 For how do you know, O wife, whether you will save your husband? Or how do you know, O husband, whether you will save your wife? 17 But as God has distributed to each one, as the Lord has called each one, so let him walk. And so I ordain in all the churches. (NKJV)
Answering your question regarding 2 Cor 6:14
Now with regards to 2 Cor 6:14
, it's under a completely different context, Paul wasn't talking about marriage here. He was saying that we don't do what unbelievers do because we are the subject of the New Covenant and we are temples of God, God is dwelling in us (verse 16) so we should obey God's commandments under the New Covenant and not act like the not-yet believers who entirely live according to the desire of the flesh.
In the context of marriage with an unbeliever - this means we don't compromise our faith and our deeds. We inevitably would interact with unbelievers but that would not change our identity as Christians.
20 I have been crucified with Christ; it is no longer I who live, but Christ lives in me; and the life which I now live in the flesh I live by faith in the Son of God, who loved me and gave Himself for me. 21 I do not set aside the grace of God; for if righteousness comes through the law, then Christ died in vain.” (Galatians 2:20-21 NKJV)
On the contrarily, we need to clearly show our identity to the unbelievers around us by living a live led by the Spirit. In John 13:33-35
Jesus gave his new commandment:
33 Little children, I shall be with you a little while longer. You will seek Me; and as I said to the Jews, ‘Where I am going, you cannot come,’ so now I say to you. 34 A new commandment I give to you, that you love one another; as I have loved you, that you also love one another. 35 By this all will know that you are My disciples, if you have love for one another.” (John 13:33-35 NKJV)
Context is important
An important principle to read the scripture is to read within the context, you can easily find passages from different parts of the scripture and come up with incorrect assumptions. Example:
5 ...Then he went away and hanged himself. (Matthew 27:5 NIV)
37 ...“Go and do likewise.” (Luke 10:37 NIV)
Now jokes aside, I would really endorse reading a whole chapter or even a few chapters in some cases to grasp what the author of the book / letter intended to communicate.
Hope that answers your question and God bless you! I pray that the Holy Spirit keeps giving us the right understanding at the right time. In Jesus Name, Amen!