"The judge starts the trial."
Translation:O juiz começa o julgamento.
I believe it could, but not very good.
O processo is more like the entire process, including the papers, containing all sessions/hearings, everything.
So the process starts when someone decides to sue another one.
O julgamento is more like the decision of the judge or the process of deciding. So, "o julgamento" is a part of "o processo".
Here you have the explanation for it (in Portuguese): http://www.blogdogramaticando.com/2014/02/juiz-ou-juiz-juiza-ou-juiza-quando.html
Is it possible to have a word with two accents?
Yes, but it is not so common. For instance, you have the word "bênção" (blessing)
And why do some monosyllabic words like já and chá take accents, when there is no doubt as to which syllable should be stressed?
It happens because they are "strongly" pronounced in a sentence. For instance: "Preciso de uma pá". Here, you have three monosyllabic words, but only "pá" is stressed.