Heather, Excellent questions. Not sure I have all the answers, but I will provide my opinions:
1. If my children were younger, yes I would immunize them with the HPV vaccine. HPV is the most common KNOWN factor leading to certain types of cervical cancer, and potentially some forms of throat cancer as well. While it cannot prevent against all forms of cervical cancer, the evidence suggests that immunizing before ANY sexual activity is most effective. So I would immunize my children (girls and boys) at a young age.
In general, I believe that once kids are old enough to understand not only short term implications of an action (this will hurt) but longer term implications of ANY health care intervention, they should be involved in the decision making. This is called patient ASSENT as opposed to CONSENT. We should try to obtain their agreement in whatever we are going to do to/for them, and immunizations are no exceptions. However, it is sometimes difficult to assess exactly how much a child at any age can understand, and so we can't automatically go with their decision.
2. Delayed vaccinations are very controversial. Many infections are most problematic for the youngest of children. Take pertussis, for example. The pertussis vaccine does not provide complete immunity until the entire series has been given, but having even one or two shots of the series can significantly decrease the severity of the disease. Very young infants are most at risk for the worst of the symptoms, such as apnea (failure to take a breath) and therefore most likely to end up in an ICU and on mechanical ventilation. So its definitely a trade-off. Pertussis is one of those diseases, like measles that is making a come back, due to falling levels of immunization in the general population. So older people may contract the disease, have a mild cough or some wheezing, and transmit it to an at-risk infant, who could potentially die from it. Over my career I have seen three infants die of pertussis, and countless others suffer through an ICU admission.
In terms of administering each component of a combined vaccine separately, that also has pluses and minuses. More separate shots are more painful (or at least your child will have to have more sticks) and will end up being more expensive, since there will be additional visits to the office and additional charges. Some of the components are hard to obtain separately these days, and the pediatrician may not have them immediately on hand. This might be especially the case for children who depend upon the Health Department (those on medical assistance or without insurance), since they can get the "routine" combination vaccines at no cost. The benefit is obviously if you are looking for side effects, and you have given only one component of a vaccine you can better assess which portion has caused the problem for your particular child. This is a situation that should be discussed with the child's health care provider, and an individual plan made that is acceptable to the physician and the parents and appropriate for the child's condition and risks.
3. This one is very hard for me to answer. Clearly, we would like to see kids get vaccines for the most common or most dangerous infections as a priority. The problem is that which infections fall into that category can change with time or with region of the country, as well as internationally. Polio is not a commonly seen disease any more, but is not eradicated (like smallpox) so there is still a small risk of coming in contact with it. A child with special needs, or who has intrinsic weakness could be completely devastated with permanent paralysis by polio, and so I cannot go "on the record" as advising that it would be OK to skip or significantly delay. On the other hand, the oral polio vaccine is often delayed because it is a live "attenuated" virus, and if the child has any kind of significant immune compromise (such as HIV) it is not given. In that case, the shot (containing the killed virus) is substituted. It is safer in that particular patient population, even though slightly less effective. Again, my suggestion is to have a serious discussion with your child's pediatrician.
If your care provider is unwilling or unable to engage in this discussion, find another provider who is more knowledgeable or accommodating.