What are the requirements for marrying someone that is in the US illegally? What rules must be followed and for how long?

Marrying just to convey citizenship or allow an undocumented alien toi stay is fraud and very illegal. I don't know if there are any published guidelines on it (probably not). But the couple would, at least, have to live together for multiple years to satisfy the authorities.

It depends on situation of each case and how the alien entered US (with or without inspection), but all must be based on a fact that both enter marriage in good faith, NOT for green card. Consult experienced immigration attorney first...

To get married, you merely get a marriage license and get married, they may still have to leave the country to apply for visa to be here legally.