Hi Oatmeal’s mama! It’s easy to become confused, there so much conflicting info about suitable vegs and greens for rabbits. The House Rabbit Society is a trusted source though, so many of us (most of us, actually) go by their suggestions. Here’s a good article about vegs and fruit for rabbits: http://rabbit.org/suggested-vegetables-and-fruits-for-a-rabbit-diet/
Rabbits can absolutely eat lettuce, it’s only iceberg lettuce that is ill reputed. Romaine lettuce is a go-to green for most of us, it’s well tolerated by most buns and it’s often available all year round.
Broccoli and other cabbage plants can cause gas in some individuals. You should introduce all new vegs slowly, to allow the tummy to get used to them. It’s clever to plan your own dinner to include the same vegs as the bun’s having, otherwise you’ll waste good vegs. A thumbnail piece of broccoli is a good starter portion.
Celery stick doesn’t kill rabbits. I cut it up in small pieces (half inch) because of the long threads that could potentially get stuck between teeth etc. I don’t know if that’s really necessary, but I still do it just to feel safe ? As with all other new food, do the introduction slowly.