A: You pose an interesting question. Is there any particular reason as to why the son does not want to become a U.S. citizen?
Derivative citizenship is very complex and case specific. However, as a general rule a minor, under 18 years of age, automatically acquires U.S. citizenship if (1) the minor is a greencard holder, (2) at least one parent becomes a naturalized U.S. citizen, and (3) the child is residing in the USA in the physical custody of the U.S. citizen parent.
There are many, many benefits to becoming a U.S. citizen. You and your son should consult with an experienced immigration attorney to discuss your son's options.
A: A refiled application should be filed as if it is being filed for the first time with all of the required initial evidence, including a valid medical exam and new filing fees. Keep in mind, if your application was denied once, it is likely that you will be placed in removal proceedings soon, as that is the standard procedure (at least here in the Carolinas). In addition, the new public charge rule takes effect on February 24th and will make the greencard process much more complex.
You should contact an experienced immigration attorney as soon as possible to discuss your case.
A: To remove the conditions on your permanent residence, petitioners must prove that they did not get married to evade immigration laws. While living apart from a spouse due to employment or educational opportunities and/or obligations certainly will make that more difficult, it is a factor that can be overcome.
You should contact an experienced attorney for legal assistance and advice specific to your case, as these type of matters can be very complex.