I'm off work because of an accident. I'm in recovery at home and have plenty of spare time. Is it good if I constantly improve my flexibility with different exercises in different times throughout the day or should I do it once a day? I wanna practice ballet and I really want to take advantage of my time. I would do exercises that don't interfere with my recovery though.
No, it's not the best idea to train flexibility continuously throughout the day.
Stretching, done properly, can be as much stress on your muscles and tendons as a cardio or light weightlifting workout. As such, you need to treat it as you would any other workout, i.e. make sure you have proper form, allow enough time for rest and recovery between sessions, proper nutrition and hydration.
As you mention you have aspirations for ballet, that is a sport that requires extreme ranges of flexibility. Due to that, I would suggest that you schedule at least a few sessions with a good ballet instructor/trainer, and get a list of recommended stretches. This will do you a lot more good than just random stretching.
Also, since you say you are recovering from injury, make sure that your PT or whoever is monitoring your recovery knows about your stretching and approves.