The correct answer is "Vinblastine-induced neuropathy." 

This is the most likely explanation for this patient's peripheral neuropathy, as the patient is described as being healthy aside from Hodgkin's lymphoma, and on no medications aside from his chemotherapy regimen. As a result, diabetic neuropathy and selective serotonin reuptake inhibitor discontinuation syndrome can each be ruled out as etiologies. Similarly, given that he is otherwise healthy and not complaining of any additional neurological symptoms (and given that he is young and actively receiving chemotherapy) it is unlikely that he is experiencing posterior spinal cord degeneration. 

Thus, in a young patient with lymphoma who is receiving chemotherapy and experiencing peripheral neuropathy, the most likely etiology is one of the chemotherapy medications that he is receiving. In this patient's case, he is receiving doxorubicin, bleomycin, vinblastine, and dacarbazine. Of these medications, vinblastine is the most strongly associated with peripheral neuropathy, as this is a fairly common side effect of vinblastine and other agents in the same class (vinca alkaloids). As such, vinblastine-induced neuropathy is the most likely etiology.

The correct answer is "PET-CT." This is the correct answer because a PET-CT shows not only spatial information from the serial slices taken on CT imaging, which would include the size and location of any potential tumor, scarring, or nodal involvement, but also the FDG-avidity or uptake of the contents of the image, which would help correlate the spatial findings with possible disease activity. Nodules observed on a CT scan can be non-descript, ranging from malignant, to indeterminate, to benign (which may still be infectious or inflammatory). The addition of PET to a CT allows the observer to see which nodules, nodes, or lesions are more likely to be indicative of disease burden, as tumors and infections will have greater FDG uptake since they are more metabolically active than surrounding tissues. 

MRI is not typically used to assess treatment in lung cancer as CT coupled with PET is more commonly used. X-ray does not provide enough resolution to adequately assess response to cancer treatment. Pulmonary function tests, while a useful measure of lung functionality, do not give a specific picture of the lung cancer burden and as such are not the best treatment response assessment tool.

The correct answer is "MRI Head." This is the correct answer because in evaluating brain parenchyma, which is relatively soft tissue, MRI provides the greatest degree of spatial resolution, which would best allow assessment of any residual, recurrent, or new tumor burden within the brain. CT imaging is commonly used to assess the brain when evaluating for bleeds or fractures, among other conditions, as it is far more expedient than MRI. However, it does not visualize the soft tissue of the brain parenchyma as well as MRI, and treatment response MRI scans are often not emergent, and therefore MRI would be preferred. PET-CT, while often highly useful in oncologic assessments is of limited value in evaluating the brain for cancer response, because the entire brain is highly metabolically active in nearly all patients, and therefore the entire brain typically has a great deal of FDG uptake uniformly, limiting the value of the scan.