Can an altered gait stiffen the plantar fascia and achilles tendon?

Compensation. You are describing the phenomena known as compensation. Any event that alters one's gait can have an effect on many areas of the foot and leg, including the plantar fascia and Achilles tendon. This is why determining the cause of foot and leg pain can sometimes be both frustrating and difficult. The compensatory symptoms can over shadow the initial problem(s), causing frustration in treatment.