Congratulations! Our first FEI was an experience and a half that still gives my trainer nightmares - but all of the others have been straightforward, so I prefer to think that’s an outlier 
Technically no, shadbellies are not required. I borrowed one for my first FEI, and then purchased one used however, as most people do wear them, at least for the CCI-longs.
Our biggest oops at the first FEI was not checking ride times after the jogs - that nearly got one of us eliminated (there were three of us running our first CCI long together). This has never been an issue at subsequent CCIs, but that first one shifted ride times by a couple hours after the jogs, so definitely double check.
Make sure you have your passport completed correctly well ahead of time, and be sure you understand the entry process - you will need to do the FEI entry as well as the regular horse show entry.
There are some minor differences in some of the equipment rules and such - do double check the FEI rules and make sure everything you use is legal. FEI is also where you’ll have to remove hind boots for inspection if you use them in show jumping, so make sure you have ground help if that’s something you’ll need to worry about. Also ground help for the vet box is very helpful.
Be very careful leading up to the event with what you feed, what liniments you use, etc. - you definitely don’t want an accidental drug test from the ‘wrong’ liniment, topical, treat, etc. Be aware of cross-contamination with other horses’ buckets if they are stacked for feeding, etc.
And remember to have fun 