I know there is no such thing as a dumb question but this is: Can you serve contextual based ads via adsense or others on a site that is entirely behind https?
You can but visitors will receive a message that your site contains secure and none secure data in internet explorer. Most other browsers have a way to communicate this state aswell.
Yes, as long as the website is not protected by SSL certificates (in that the client accessing the website needs to have a certificate) Google is perfectly capable of indexing your site for keywords to cater the right ads for your website.
I am not sure if Google makes the adsense code available over SSL as well, if not your visitors will be warned by the browser that the page may contain insecure elements. I do know that their analytics code (For Google analytics) does contain an SSL possibility.
In case you have any more questions, Google's adsense support team/faq will be able to better provide answers to your questions, since you can let them know what site you are talking about!
Google is aware of the issue but does not offer a properly configured HTTPS version of their code at this time. It'll work by swapping out http for https, but as mentioned above you'll get various errors in browsers.