If they're in the country legally, they should apply for US citizenship.
Yes. If the illegal immigrant marries a citizen, they can become a citizen from that.
No, marrying a US citizen or a permanent resident does not grant the illegal immigrant legal status regardless of whether there is a child involved or not.
Check with the laws in your state before getting married, but in most states, anyone who marries a United States citizen can become a US citizen, and thus be legal.
No, she is not. She will be allowed to remain in the US with her spouse and children, but she will have to apply for citizenship separate from her marriage. The US citizen REMAINS a US citizen. The immigrant remains an immigrant and must follow the legal path to resident immigrant status, and from there to citizenship. Marriage is not a free pass to residency or citizenship.
Yes, they could think the marriage is just to stay in the country.
It is legal in the United States for an illegal immigrant to marry a US citizen. Once the marriage is legal, the illegal immigrant becomes a legal immigrant and can stay in the US for the course of the marriage.
no
no
No.
If they have a green card, they can work.
No, but it does give you the ability to complete the naturalization process.
The process is called "naturalization" - the legal process by which a foreign immigrant gains citizen status, and (almost) all rights and privileges thereof. Naturalized citizens still cannot be President or Vice President.