I dont think the medicine ball sit up is the best exercise for our situation as it also works out the hip flexors. For core I would work on planks and their variations. Also, glute bridges target the core as well. Even chin ups/pull ups work the core intensely if you have access to a chin up bar. Try to find exercise which dont involve hip flexion for core activation. 

You mention squats cause you pain. Try doing wide stance box squats. Its a very good exercise for posterior chain. It targets the Glutes, and hamstrings heavily and helps teach proper squatting form. It will make the transition into normal squats a lot more easier. It will teach you how to squat using your hips, glutes and hamstrings rather than your quads. After, when you progress to normal squats, focus on preventing the knees from tracking over the toes. Use your hips, glutes, and hamstrings rather than your quads to squat down. Pushing your knees out during the squat helps keep your knees from tracking over the toes. If your knees track over the toes it puts a lot of stress on the quads and patella.