If I have soreness and pain in the back of my calf that makes me limp when I walk, is that a torn muscle?

Tender to touch? Medial gastrocenemius or popliteus tendon local tenderness is a hallmark of both but DVT (clot)can be tender as well.
Depends. Occasionally this can represent claudication which is a pain in the calf muscles when you walk a certain distance. This is due to a lack of circulation to active muscle groups in the calf. This usually stops after a few minutes of rest.
Calf Pain. First, is your pain related to the length of your walk or does it occur right away. Does it involve both legs?If it happens after you walk for some distance it can be exertional compartment syndrome. This occurs when the muscle lining, the fascia, restricts the calf muscle. Other causes are circulation which can cause pain. I doubt you have a torn muscle unless you recall an injury.
Maybe. You could also have poor arterial blood flow into the calf, also known as peripheral arterial disease (pad). Blockages in the arteries that lead to your leg, or are in your leg, will decrease the blood flow to the lower leg. At rest your calf muscle does not need much oxygen or blood flow, but when you walk the muscle needs more blood flow, and it will hurt or feel sore when it can't get enough.
Possibly. Do you have recollection of an injury or did this come on acutely? If so, it likely represents a strain/partial tear of one the calf muscles- the gastrocnemius, soleus or plantaris. The most common calf injury is a partial tear of the medial (inside) gastroc. We treat with rest, ice, a heel lift and sometimes crutches and/or a boot if severe. A blood clot has to be considered if no injury.