Is it permisible to have temporary marriage with a widow? If so, what are the conditions?

How should the nikkah form be, public or secret? What if the woman wants it secret? Or the society we live in do not understand this concept

Who are those with whom to do nikkah muta?

As per Shia 12 Imami school of thought there is no need of witnesses during Nikah is mandatory but at the time of divorce yes it’s mandatory to have two just person. But it’s recommended (Mustahab) to have witnesses during Nikah. The most important condition for both permanent marriage and temporary marriage (Mut’a), if the girl is virgin and dependent upon her father, it is wajib (obligatory) to seek her father’s consent but if she is not dependent and not a virgin and she is either widow or divorced then she can take her decision on her own. You should also have fixed time period (how long you want to be together) and Mahr ( dowry) before Nikah.
Whom you can do Mut’a is same as with whom you can do Nikah (permanently).