I should know this by now, but are we allowed to bring in our own first aid kits, complete with trauma shears?

I'm assuming you're a first responder of some sort and I have no idea if little scissors with blunt edges violate convention policies. However I'd recommend carrying whatever first aid kit you feel comfortable with and plan to stay out of the way unless it's absolutely necessary. No doubt Gen Con and the ICC have detailed plans from a stubbed toe up to a mass casualty incident. In a lot of cases attendees trying to help when those plans are enacted can cause more harm than good.

However I absolutely do respect and appreciate your instincts to help out. 

It's more for personal use. I never intervene or interfere if I'm not on duty and there are other providers on scene.