For now, don't think too much about the signature campaign, it's better to increase your posts to increase your ranking.
1. What is the exact requirement to join a signature campaign?
At least become a full member, most BTC paid signature campaigns are acceptable ranks.
But when you become a Jr/Member rank, you can wear a signature but the text is limited.
2. How do I increase my rank from Brand New to Newbie?
You are now a newbie.
Continue to increase your posts with quality... later merit will come by itself from the posts you make, it can increase your account rank.
3. How many posts or how much time does it usually take to become eligible?
You are free to post as much as you want, as long as you don't spam.
Newbies usually have to wait 360 seconds for each post if they want to post.
4. Are there any official threads or guides I should read first?
Yeah, you can read the link guide that has been listed by several users.