Some rabbits are sturdier than others. It's like when you meet people who tell you their great uncle lived 100 years while smoking a pack of cigarettes a day and drinking heavily. It happens. If Aki goes without eating as much hay as usual for 48h she'll go into stasis. Everytime without fail. Most rabbits also get horrible teeth problems if they go without hay. Your previous rabbit was super lucky and had genes of steel, I guess.
Still, what Blue eyes said is completely correct. Carrot and apples aren't things a rabbit should eat everyday and they don't count as a vegetable portion, they are treats. A rabbit should eat about 8% of his body weight in vegetables everyday, with at least 4 different vegetables (changing with the seasons, obviously) and about 2/3 of leafy greens and 1/3 of hard vegetables. Pellets should be around 2 to 3% of the rabbit's body weight. To answer your original question, rabbits can (should) eat some lettuce every day. Batavia, chicory, curly endives are all staple foods. Carrot tops are a lot better than the carrot in itself but should be given more sparingly than some other leafy greens because they have a lot of calcium in them (same : some rabbits will get bladder stones and UTI really easily, that's not a problem I've ever encountered with my rabbits but I know people who have to be really careful with the calcium thing).