Ninety percent of muscle injuries are either a muscle strains (microscopic tears in the muscle) or contusions (bruising of the muscle). A muscle strain typically occurs at the point where the muscle and tendon attach to each other and is the result of excessive force on the muscle and tendon.
The good news is that muscles have a good blood supply and heal well. The not-so-good news is that the muscle will heal with a scar that won't contract, so it needs to be protected, especially immediately after injury by limiting your weight bearing on the foot by using crutches or some other type of assistive device, or by wearing a walking boot. Depending on the extent of the tear, you may be immobilized from 10 days to 4 weeks. Research evidence reveals that early movement can minimize the impact of the scar. After a thorough evaluation, your physical therapist can address any loss of range of motion and changes to the soft tissue prior to beginning muscle strengthening and functional training like walking. As you transition from complete immobilization to no immobilization, you may experience some discomfort as you increase the amount of stress applied to the muscle. So although I tell my patients to "respect" pain, it is not the sole indicator for stopping movement. Be sure to communicate your comfort level with your physical therapist.
I would expect your foot to be stiff after a long night of sleep. The foot is not moving so the blood flow is decreased (this is normal), so the tissue is less hydrated and stiff. If you massage your foot or pump your foot like you're pressing and releasing the gas pedal of a car prior to stepping out of bed, this will decrease the stiffness and allow you to walk normally. Your physical therapist can give you other strategies to reduce or prevent the feeling of stiffness in the foot.