Can someone die of the common cold?
The common cold is a clinical syndrome restricted to upper respiratory tract involvement. By clinical syndrome, I mean it is the constellation of symptoms (rather than the consequence of a specific pathogen). As you mention, these symptoms are the result of the immune response, rather than tissue damage or compromised function as a direct effect of a pathogen or its toxin (e.g., the watery diarrhea in cholera).
As defined (see, e.g., Cecil Medicine Ch. 369), this clinical syndrome cannot lead to death.
The common cold is an upper respiratory syndrome of rhinorrhea and nasal obstruction, frequently accompanied by sore throat, sneezing, and cough.
Can the viruses that cause the common cold cause death?
Yes. Many viruses that cause the common cold also cause other clinical syndromes that can cause death. This occurs when viral replication moves to the lower respiratory tract. As an example, influenza viruses are responsible for 25 - 30% of common colds (see Bennett, Principles and Practice of Infectious Disease, Ch.58). When it moves beyond the upper respiratory tract, influenza is responsible for substantial mortality. Other virus families that are responsible for both a common cold syndrome and lower respiratory tract syndrome in immunocompetent individuals (e.g., bronchiolitis, pneumonia) include parainfluenza virus, metapneumovirus, adenovirus, and (rarely) coronavirus. Rhinovirus, responsible for 40-50% of common cold cases, is uniquely unsuited to lower respiratory tract involvement, because of its preference for the cooler environment of the nasal mucosa, replicating best at 33 C (Murray Medical Microbiology, Ch 56). However, in individuals with Severe Combined Immunodeficiency (SCID), lower respiratory tract involvement does occur. There are a number of case series reporting death due to lower respiratory rhinovirus. This is an example.
Can the common cold lead to serious illnesses other than lower respiratory tract involvement?
Yes. Other morbidity can occur as a result of the immune response that produces the common cold syndrome. Rhinorrhea and congestion can progress to a viral rhinosinusitis, a separate syndrome with its own complications, or a secondary bacterial infection, which can lead to bacterial sinus involvement and/or bacterial lower respiratory tract infection. Otitis media is another common complication, especially in children, and has its own potential complications. Asthma (and, generally speaking, most lung diseases) can also be exacerbated by what would otherwise be a simple common cold and predisposes to lower respiratory tract involvement. Asthma does deserve special mention, because rhinovirus associated exacerbations can be fatal, but this is a consequence of asthma rather than a common cold syndrome. Further discussion of these syndromes are beyond the scope of the question, but are discussed briefly in the chapters referenced above.