For fitness, if your horse is not in great shape I wouldn't reccomend starting with gallop sets! That can do more damage than help. The most important thing about fitness training is listening to your horse- how he is breathing, tiredness, etc. I would reccomend starting with trot sets before cantering even. What I did with my pony was 3 minutes of trotting, then 1 minute walking, and repeat this 4 times. If your horse finds this to be easy, increase the duration of trot, but if he finds it taxing, don't be afraid to give him another minute walking! Conditioning about twice a week is usually enough because if your horse is out of shape, overexertion can do more harm than good.
THIS! You shouldn't need gallop sets until at least Training. If you are riding 4-6 days/week for an hour of proper work, you should be fine. Another good rule of thumb is the 3x4. That means 3 sets of 4 minutes at competition speed (yeah, you'll want to start learning about pacing and speeds). You work up to that slowly with trot sets and gradually increasing the canter. Increase duration or intensity, but never both.
For dressage, you definitely don't need a dressage saddle! I know people who go training and don't have one. ;) Braiding really isn't need if it's a schooling show, but if it's recognized I feel like braiding just looks better. "Faster" transitions aren't usually better; it's more along the lines of "correct". As in, the horse is using his back, rider half halts to balance, etc.
This. Check your omnibus to make sure it says small arena (20x40m) and be prepared for either grass or sand. Accuracy, rhythm, and balance are the most important things to practice. I would NOT buy an all-purpose saddle. They aren't right for either phase. Use your jump saddle for dressage and save your money for a dressage saddle. You can easily get to Prelim without a dressage saddle. Agree about braiding. It shows respect for the sport and while you will not be directly docked for not doing it, braids make a good impression--and we all know first impressions matter.
Cards for medical armband can be found on USEA and USPC websites. I find Woof boots are best for XC, but it's pretty much a personal opinion. Again, for stadium attire it depends if it's recognized. If it is- you should probably wear a coat, if it's not your xc polo is fine.
A few things here: I, too, like the Woof boots. I don't care for SMBs because of the potential fetlock restriction, but it's about personal preference. The "rules" for boots are that it can't be problematic when wet, not too hot, and nothing with fleece that could catch burrs and the like. ABSOLUTELY NO POLOS.
SJ attire--plan on the coat unless they wave them for heat, in which case you wear your white or light collared shirt without the stock tie.
Most people talk about XC polos, but you don't actually have to wear a polo for XC. Nobody can tell under that vest! I wear a sports t-shirt (short or long sleeved depending on weather) at Training level. I never understood the obsession with the XC polo, but that's me. I hate collars and things on my neck, and it's the last thing I do up before dressage/SJ. Personal thing.
Also, you really don't need studs until Novice, so you should be good :) For dressage, just practice the movements in your test. Don't repeat the test too many times though because the horse can start to anticipate. For example, if your test says halt at X, turn left. Halt at X, and practice turning both left and right.
WOW that's a lot.... Sorry 'bout that ... Good luck!!! :)
Unless your horse is super prone to slipping and/or the footing is beyond horrendous you shouldn't need studs even at Novice. I know some events (Rocking Horse in Florida) have such good footing that nobody uses studs, even at top levels. You won't need them.
Good advice on the dressage test. Practice parts of it out of order, practice the other test, practice the mirror image--but don't let the horse learn the test (they do that remarkably quickly).
Be sure to read your rule book and know things like legal spurs, warm up rules, etc. Jumps are flagged in warm up. No standing martingales. No coaches riding the horses. Horse must wear number when not in a stall. That kind of stuff.
I'd go watch an event or even volunteer. You'll learn so much about what to expect. Things like horses spooking at someone in the judge's box, the routines for bit check, how a start box works, etc.
Best of luck!