I use the bead jack on my carbon rims, can confirm no problem.
I use one of the bottle cage storage bottle things and normally carry:
Tube
Co2 inflator with 1 cartridge
Old quick link and a piece of brake cable
Cash, $10-20
Multi-tool
I even manage to fit my car keys and airpod case in there
For multi tools I would suggest one with as many functions as possible. I use a Topeak (Hexus? I think) that includes allen keys up to 8mm, tire levers, valve core remover, chain breaker, chain hook thingy, torx, etc. They're actually quite small and has more functions than I'll ever need, and they make some with even more finctions. Strongly suggest not omitting tire levers, roadside repairs without them are no fun when you're tired, especially on super tight tires
I sometimes bring a drawstring backpack and shove all that stuff in it plus more if I'm going for a long ride or plan on making stops. Honestly, I barely notice it on my back and I prefer that to shoving a bunch of things in my pockets