No, veggies won’t make a rabbit fat. Veggies are low in calories and are mostly water. Pellets will make them fat, and I suppose it’s possible that veggies could make a rabbit chubby if it was disposed to be chubby (and very inactive), but a normal active rabbit won’t get fat on veggies. Mine get at least 2 cups a day (each) and aren’t fat. They get very limited pellets.
All of the veggies you listed are just fine. Some rabbits are very prone to bladder sludge or stones and a vet might recommend to limit certain higher calcium veg (like spinach and some greens) but most rabbits can eat them just fine. Lettuce, parsley, and celery aren’t high in calcium.
Most of us recommend a variety of veggies just because variety is always healthy. However, they don’t *have* to have a variety. One of my rabbits is picky and can have a sensitive tummy, so they eat romaine, leaf lettuce, and parsley every day and are fine. Sometimes I’ll feed the other bun some different veggies that I’m cooking with since she likes new things and doesn’t have tummy issues, but that’s a very small part of her diet. If your rabbit likes a variety, maybe pick a new type of veggie every week when you go shopping (you can keep lettuce as the base and maybe try different herbs or different greens mixed with it).