Dentist had to do fillings and my cheek was very numb but my tongue and lips not so much. I still felt the work being done. Normal?

Talk to your dentist. My recommendation would be to discuss your concerns and experience with your dentist. When getting any fillings done you should not experience any sensitivity or discomfort.
You needed more. You needed more local anesthetic. Sometimes, especially for lower teeth, one application of local anesthetic is not enough. In the future if you feel you are not adequately numb you must inform your dentist. They can easily add more local anesthetic. There is no reason to feel any pain during treatment.
Not really! Depending on the type of procedure needed for a patient there are different types of local or block injections. Block injections, which are more profound, and numb up a bigger area are normally used for back teeth procedures as well as surgical procedures. Local injections normally used for front teeth and smaller procedures. The Key is to always give feedback to your dentist during procedures.
Anesthesia. It appears that all of the nerves that your dentist was trying to numb were not numb. Therefore it is normal that you felt discomfort while being treated.