The judges were not a part of the monarchy; they were a separate institution. There function was, of course, to judge disputes.
It is the book of Judges.
Eli and Samuel were the last two judges of Israel before the era of the monarchy began.
Elijah
no
The prophet Samuel served as a bridge between the Judges and the Kings of Israel. He anointed both Saul and David as the first two kings of Israel, transitioning the nation from a period of judges to a monarchy.
Deborah (Judges ch.4).
No, Barack is not mentioned in the Book of Judges in the Bible. The Book of Judges primarily focuses on the period of time between the conquest of Canaan and the establishment of the monarchy in ancient Israel.
There was never any place called "Hebrew". If you are talking about Israel, it was a monarchy in ancient times. Today it is a parlimentary republic.
No. They are meant to be 'above' politics - today we might say 'outside' politics. That means they are unbiased judges, who do not stand to gain from the fortunes of any party or policy. Their function is to advise and to warn. The institution of monarchy is easy to ridicule. But many republics have seen the wisdom of restoring the monarchy.
The Hebrew word "oht" (אות) is sometimes used in that context (Judges 6:17).
According to the Hebrew Bible as it is found in today's printed editions, Eichah (Lamentations) is after Ruth.
No, that is not a function of that part of society.