Everything the books tell us of the Free Companionship can be found at this link.
And based on what is complied there, I would say yes, in Torvaldsland, free companionships are done by contract and being contracts, are renewed each year.
Did free companions refer to each other as "Mate"?
I find no evidence of the word "Mate" being used as a pronoun.
Was the word "Mate" used to refer to a Free Companion?
Yes, in several references, the word "Mate" is used to refer to a Free Companion.
Here are all references to the word Mate as it pertains to one in a Companionship.
"She agreed to be the mate of Pa-Kur, the Assassin," he said, "in order that you might have one small chance of life, on the Frame of Humiliation."
Tarnsman of Gor    Book 1    Page 159
"Perhaps she is someone's mate?" I asked.
"No," said the first slave.
"How do you know?" I asked.
"She is not in the breeding cases," said the man.
Priest-Kings of Gor    Book 3    Page 103
Similarly a woman who was particularly ugly, threatening, vicious, stupid, cruel, etc., would not be a desirable mate. No man can be blamed for not wishing to make his life miserable.
Hunters of Gor    Book 8    Page 312
To see a girl's lips and teeth is a charged experience. To touch a girl's teeth with your teeth is prelude to the seizure of her body, an act that one would engage in only with a bold, brazen mate, or with one's shameless slave girl, with whom one can do with, to her joy, precisely as one pleases.
Tribesmen of Gor    Book 10    Page 213
I would later learn that the unmated daughters of even important men, namely, good hunters, were often kept in the poorest of furs. It is up to the mate, or husband, if you wish, to bring them good furs.
Beasts of Gor    Book 12    Page 193
In other tribes, such a man, if he is over twenty-five, is permitted to mate, but he is not allowed to paint his mate's face. Thus will her shame before the other women be made clear.
Savages of Gor    Book 17    Page 45
That he had painted her face was also impressive. Usually, among the Kaiila, it is free women who are permitted face paint, and then, commonly only at times of great festivals. This paint is commonly applied by the woman's mate.
Blood Brothers of Gor    Book 18    Page 118
