Is a creaky foot a sign of tendon and ligament injury?

Could be. Without a full exam, this is difficult to get a complete picture. Some people will hear and feel popping from tendon injury but others will experience similar symptoms from joint injuries. This is probably something that you want to have evaluated by a foot and ankle specialist (podiatrist or foot/ankle orthopedist).
Depends. It is common for joints to make noises. Is the creaking noise painful? Is the noise associated with an injury or loss of function? Is there swelling? The bottom line-if the creaky noise is affecting you function please get it evaluated.