To provide a brief answer, as I think a sufficient answer would be far beyond the scope of this site, the symptoms that are "generic" to the common colds are all caused by the body's response to a pathogen (it's why they are generic). Specific pathogens tend to cause additional symptoms on top of that, some of which seem to be directly related to viral damage.
First, I want to point out to a very nice website/data set from the Common Cold Project (CCP). The publications from the comprising trails is a good repository for information on the common cold.
*It should be noted that there were a few experimental design problems that they were unable to avoid (for a combination of cost and lack of knowledge). It turned out the the RSV challenge strain used in the British study was not a very pathogenic strain. Further, as it has delayed onset of symptoms meant that it wasn't well studied with in the time frame of the trial. That said, it provides a good deal of details on HRV, and the "common cold" more generally.
What are the common symptoms of the common cold?
This might seem like a more obvious question, but it's what must be answered first before the broader question of what is being caused by the virus is addressed.
One of the better lists of symptoms comes from the first British study mentioned in the CCP:
- Sore throat
- Nasal obstruction
- Increased tissue usage
- Nasal stuffiness
- Sinus Pain
- Post-nasal discharge
- Cervical adenitis
- Extra bed rest
- Watering Eyes
- Myalgia [muscle pain]
- Mucopurulent discharge
- Temperature increase in the day
- Temperature increase at night
[Table 3, ibid]
Not a single one of these is actually caused "directly" by the virus, but it kind of get's back to what can the virus do? Viruses are very small obligate parasites that rely on changing host systems to replicate and spread their genetic material.
If we take a look at the snotty, cough, and sneezy symptoms (cough, bronchoconstriction and airway mucus secretion), they are all "caused" by rapidly adapting receptors (RARs) along the airway. That is to say, the damage the viruses do actually cause to cells along the airway is detected by RARs either through the cell debris, chemical signaling (indirect activation from other cells), or direct interaction of immune cells.
If we look at a runny and stuffed up nose (rhinorrhea and nasal obstruction) specifically, we know that neutrophils are coming to the sinuses and inducing inflammation and mucus over production.
What do the viruses actually do then?
This is where the question gets very broad very quickly. I will deal with three viruses as examples: HRV, RSV, and influenza. Some of the best sentinel data on colds, at least colds severe enough to seek medical treatment, has been coming from China. The two leading candidates from these studies are HRV and RSV (also true in US, see a related SE.BIO awnser of mine). I want add in flu because it has some other interesting traits.
Both HRV and RSV often are found with co-infection of other viruses or bacteria, but RSV is more likely to have this problem [ibid]. While "drained" resources while fighting the common infection might be a partial explanation for this, we know that RSV produces two proteins, NS1 and NS2, which directly inhibit the innate immune system. This could explain the high rate of co-infection with RSV. Which leads me to the first general thing that viruses do, directly counter the immune system of a host.
Another comparison would be what kind of direct cell damage (cytopathology) the virus might do. RSV and flu tends to destroy the airway epithelial barrier by infecting and killing the epithelial cells (this sets of the RARs). HRV on the other hand, doesn't often kill the cells off, but just disrupts their function [ibid]. Thus the second general thing viruses do is "kill" or "disrupt" the cells they infect.
The final action of a virus I want to look at goes back to a symptom that is on our list, myalgia. Of our three viruses, this is most closely associated with flu. Though the exact mechanisms may vary, it is believed to be caused cytokine release.
Here's a good example where answering the question of what is "caused" by the virus is tricky. Some influenza strains produce proteins where a main function seems to be inducing cytokines. This can lead to all kinds of bad things like a cytokine storm. But flu also needs to cause an up regulation of trypsin to help process it's proteins. This might be done indirectly through a cytokine or directly from a viral protein. Either way the trypsin can remain in the tissue causing damage to host proteins after the virus has moved on (been cleared). So would we say this was tissue damage caused by the virus? It wasn't an immune response in the normal sense, and it was started by the virus, but then again the trypsin is not a viral protein.
That leads to my final overly generic statement about viruses, they activate complicated pathways with all kinds of incidental downstream consequences (normally bad). Imagine if someone was in front of a power breaker box randomly turning off and on switches. If the food was spoiled in the refrigerator would you say it was because the refrigerator wasn't on, or because somebody pulled the fuse out?