Ok - at this time if he does not have proof of legal entry, Cancellation of Removal will be his only option. It requires that he have ten years here, good moral character, and that a very very very high level of hardship would result to either US citizen children or a US citizen spouse. If he were to marry his wife could file an I-130 Immigrant Visa petition for him, but he would not be able to adjust status in the US without proof of legal entry. There is an option of consular processing, but he would then need to leave the US and if he leaves now he will trigger a ten year bar to his return.

Yes, he would need to show an extreme level of hardship to the US citizen children should he be removed from the US. Ten years physical presence in the US is required, so he clearly meets that. If he would marry the mother of the children that would likely be a good idea. The I-130 filing would at least be something started, while deciding whether to apply for Cancellation of Removal.

Yes, they can marry whether he has status or not. If they do marry, she can file an immigrant visa petition for him. If he had entered legally, or - likely - if the laws change - he could then get a greencard based upon the Marriage. Because he doesn't have proof of a legal entry, the marriage based immigrant visa petition would require him to leave the US and go to a US Consulate to apply for the greencard. Leaving the US would trigger a ten year bar to his return - there may be waivers, but it would still be a long wait. So instead, I would suggest they marry and file the I-130, to either sit on it and wait for a change in law so he could apply here - or, at least, so they have a plan started should he be caught and removed from the US, or look to applying for Cancellation of Removal, if there would be issues for either the kids or his wife (to be) without him here.