how anyone can become mod of a specific board?
I think they're appointed independently, but for local staff usually theymos first ask the local community to suggest candidates who then hold a vote to determine 1 or more users.
do mod get paid? who pays to mods and how much they earn?
At every forum ad space auction event in the past, theymos mentioned that in the first paragraph.
https://bitcointalk.org/index.php?topic=920291.msg10110267#msg10110267About 25% of ad income goes to the forum moderators as thanks for all of their work. (There are many moderators, so each moderator gets only a small amount -- moderators should be seen as volunteers, not employees.)