those are a lot of questions

I've done obedience, rally, 4 different agility organizations, and disc dog.
Earning a title: For most agility, obedience, and rally organizations you need 3 qualifying scores at each level to earn a title and the ability to move up to the next level. Some organizations and specific titles require you to move up in level, other organizations have "championship style" titles within each level and allow you to stay in that level longer if you choose. In disc dog, there are no titles although most competitions have placements and medals for 1st-3 in the "novice" division and "pro" division. No one polices what division you should enter, although the general unspoken rule is once you win in novice you for sure need to move up to "pro" and most people on their 2nd dog just start in pro even though the dog might be very green itself.
Scoring: In agility most runs are based on time+ faults. Most organizations require you to have a "clean run" to qualify in that any knocked bars, off courses, missed contacts automatically NQ you. But there are a few organizations out there that you can still qualify even if there is a knocked bar. Usually placement ribbons are given to the top 4 teams, even if only the top 2 teams actually qualified, qualifying ribbons are also given out.
In obedience the scoring is based on the judge's opinion of the team as compared to perfection. You start with a perfect score of 200 and the judge gives the orders for each exercise and then takes off points for each error. Errors range from an automatic NQ for things such as a second command (except in heeling) to substantial errors of 3 points for things such as a no-sit and skipped front, to minor errors of .5-1pt for things such as minor lagging/forging, crooked sits. You need 170pts to qualify at each level and at least half the points from each exercise. Placement ribbons are given for 1st-4th place in each level and qualifying ribbons are given to everybody that qualified. In obedience, teams are also divided into "A" and "B" depending on whether the dog already has the title for that level or not or whether it is the first dog of that handler to ever compete.
In rally, the scoring is similar to obedience however things that would get .5pt off in obedience are not deducted in rally. Because of this, it's not uncommon for good teams to easily get a perfect 100pts even though there were things such as crooked sits, bumps, or minor lagging. In rally, handlers are also allowed to talk to their dogs as often as they want, where in obedience 2nd commands are either an NQ or points off depending on the exercise.
Organizations:
In obedience there are a couple different organizations. For newbies, I highly recommend
CDSP if it's in your area (your bio says NC and there are trials around there!). CDSP and AKC are very similar in their requirements, 3 Q's under 2 different judges to get each title and move up, have mostly the same exercises with a few variations (such as recall over a jump in CDSP and no jump in AKC, etc.), but CDSP is much more relaxed and allows one give a treat to their dog in between exercises. Other organizations that have obedience are UKC and ASCA with rules very similar to AKC.
In rally, APDT is my favorite option. It allows handlers to give treats at certain signs, but has longer courses compared to AKC. At the most advanced level, APDT has much more advanced signs as compared to the other organizations. It is also more relaxed than AKC.
In agility, there are 7 organizations holding trials in my area (minnesota)! All have different styles of courses and different games. I currently compete in USDAA and NADAC with my Toller, and only NADAC with my corgi although I have tried out a total of 4 of the organizations.
In Disc dog there are 2 popular organizations in this area. My local club puts on both types of trials. As there are no titles in either one, people compete for fun. There are regional qualifiers though for the national event for each organization.
About the sports:
I compete in obedience because I love the challenge of making obedience fun for my dogs. It is really the only sport that is not self reinforcing for the dogs, lasts the longest, and you only have yourself in the ring to be a reward. Many people think of obedience as boring and ridged, but it is all in how you train. I aim for high levels of drive, attitude, AND precision with my dogs. The downside to obedience is the lack of positive trainers to learn from and the snotty attitude some people hold towards positive trainers. On the positive side, Denise Fenzi has proven that getting a positive trained OTCH is possible (got it on 2 of her dogs!) and now is even setting up an
online "academy" to help others around the world.
Rally is similar to obedience but less formal. The positive to rally is that it can be used as a nice stepping stone to obedience. Rally is mainly heeling so I love doodling rally moves in my training for obedience. I mainly only compete in rally to help build positive ring experiences for my dogs. The downside to rally is that if one trains just for rally they can be doing really sloppy training. Because rally is so lenient in it's scoring, many people coax and plead their dogs through it so basically their dogs don't have the solid foundations for it and are confused and stressed in the ring. The other downside to rally is that some dogs don't like all the stopping and starting it involves. Some dogs thrive on the constant change of direction, but some dogs don't like the opportunity to never get into a rhythm with their heeling.
Agility is fun because it's so much easier to motivate your dog for 30seconds of pure running. I love it because it has very little to do with obstacle performance but is all about the HANDLING in between the obstacles. It is fast paced and an easy sport to train in a motivational manner. The downside to agility is that if you have a very drivey dog and a handler who is constantly late in their cues, it can take a great wear on the dog's body.
Disc is fun because the reward is right there

Dogs naturally into retrieving can really turn onto this sport and with the right foundation even non drivey dogs can learn to love it. The downside is that the potential for injury can be high IF the handler doesn't throw in the correct location and early enough. Any time we ask the dog to leap, especially off our body, the dog needs to be able to predict where the disc is going to be way BEFORE he leaves the ground. Too late and too low throws are a recipe for disaster.
Getting Started
Getting started in any sport I recommend having an instructor so you build the solid foundations. Foundations are crucial to keeping your dog safe, preventing having to retrain, and prevent a dog who stresses in the ring because he doesn't know what's expected or the handler is too late in their cues.
- For agility, there are many people who do online classes now if you don't have access to a good instructor. My favorite is Silvia Trkman, but Daisy Peel, and Tracy Sklenar also offer online classes.
- For obedience, Denise Fenzi is my favorite but Fanny Gott also offers online classes.
- For disc,
Pawsitive Vybe with Ron Watson is amazing and offers a ton of free videos on his site.
Before Competing:
Before competing in any sport you should make sure your dog can do it in as many locations as possible with only the things you can legally have in a trial. Distraction training is HUGE

For obedience and rally, I also put a lot of emphasis on "ring entrances." I want to make sure my dogs know the ring is a super fun place to be and that they are ready to work the second they pop in the ring.
My dog needs to be confident in new environments with lots of strange people and strange dogs.
My dog needs to be comfortable relaxing in that environment as many trials you're there for at least an hour if not several, or even all day for some agility trials. Some people crate out of their car, most bring crates in the building.
I hope some of that helps
