Can I get the same HPV virus twice?

You can never presume that HPV is not contagious. The medical community used to believe that herpes was only contagious when a person had an outbreak. They now know this is not true and that it is contagious at any time. HPV, also a virus, usually has no symptoms and if involving the vagina (for example) one would not even know they had an active lesion.
Because HPV affects far more than just the cervix, a negative pap is not the end of the story. HPV can also affect the vulva, vagina and anus. In addition studies now show it to be the major cause, even more than smoking, for oral cancers.
As far as being reinfected, since you never get rid of the virus, technically you cannot get reinfected with the same strain. HPV goes dormant but can recur weeks, months or years later. I'm glad to see your family doctor is as educated as he is regarding HPV, many are not.
A gyn-oncologist is, in my opinion, preferrable to an ob-gyn because they are more familiar with dealing in these types of precancerous lesions as an ob-gyn has a practice of pregnant women to deal with as well. Someone solely dedicated to conditions such as this would again be more preferrable but that's my opinion (and the opinion of many women who have gone through high risk lesions and cancer resulting from HPV).
You can always have more than one strains of HPV. Some could be high risk, some low risk. So you may also have a strain which causes genital warts but have not had any symtoms yet. Low risk strains of HPV can cause effects on the cervix but do not progress to cancer. It is important that you have an HPV test performed to detect any high risk strains of the virus.
Please also refer to thehpvsupportnetwork.org for more information.