**Core Concept**
The patient's presentation is suggestive of a condition affecting the thoracic aorta, leading to aortic arch syndrome. This condition is characterized by the involvement of the aortic arch, which supplies blood to the left arm, left side of the face, and left eye, among other structures.

**Why the Correct Answer is Right**
The patient's symptoms of weight loss, cough, and hemoptysis are classic for lung cancer, specifically squamous cell carcinoma of the lung. The gradual progression of left upper limb pain, paraesthesia, and weakness, along with Horner's syndrome, is suggestive of a Pancoast tumor, a type of lung cancer that arises in the apex of the lung and invades adjacent structures. The aortic arch syndrome is caused by the tumor's invasion of the aortic arch, leading to compression and ischemia of the left arm and left side of the face.

**Why Each Wrong Option is Incorrect**
**Option A:** Pneumonia would not typically present with Horner's syndrome and aortic arch syndrome.
**Option B:** Pulmonary embolism would not cause Horner's syndrome or aortic arch syndrome.
**Option C:** Mediastinal lymphoma would not typically present with Horner's syndrome and aortic arch syndrome.

**Clinical Pearl / High-Yield Fact**
Pancoast tumors are a classic example of a tumor that can cause aortic arch syndrome due to its invasion of the aortic arch. It is essential to consider this diagnosis in patients with lung cancer and symptoms suggestive of aortic arch syndrome.

**Correct Answer:** C. **Pancoast tumor**