When a proper noun like "Skype" is used as a verb ("Skyping"), should it be capitalized? My thinking is that it should be capitalized because the root is a proper noun. Does anyone know of a rule about this question?
The answer is tied up in genericized trademarks and neologism. Google is a good example of this. Google the company has fought very hard to prevent its name from becoming genericized. Around the turn of the millenium, "google" was on its way to becoming a common term for web search of all kinds and Google took significant action to try and prevent this.
The word "google" has been added (as lower case) to some dictionaries including Merriam-Webster, but as a result of Google's pressure the definition is some variation of "using Google's website to search the internet".
I haven't seen "skyping" used to describe video chat that is not from Skype itself. I'd use that as a rule. Once it's common for people to describe using FaceTime or something similar as "skyping" is when it's reached that genericized point.
Etymology is not a determiner of whether something gets capitalized or not. Many proper nouns are derived from non-proper nouns (e.g. Apple, Smith, United Kingdom, World Wide Web), and vice versa (e.g. atlas, echo, narcissist, siren, sodomy). It is usage and usage alone that determines whether something gets capitalized or not. In fact quite a few proper nouns become genericized against the owner's will because of usage. Examples of nouns created that way include aspirin, bandaid, coke, escalator, kleenex, thermos, zipper; examples of verbs include google, photoshop, facebook, and skype.
Of course when you coin an all-new verb — say, to AltaVista, to iPod, to McDonald's, to NATO —, capitalization is instrumental in helping readers understand what on Earth you're talking about at all. But once a verb has entered everybody's everyday lexicon you'll be hard pressed to justify capitalizing it, especially if it's taken on a generic meaning, and especially seeing how English, in general, simply does not capitalize verbs except at the beginning of a sentence. "I googled for photoshopped cats and had to lol" becomes the norm, not an exception.
To answer the question actually asked:
Proper nouns and words derived from them are capitalized. That's why adjectives like French are capitalized. Thus we find words like Christainize always written with capitals.
Nouns can become improper by virtue of genericiztion, and Google is probably one of those. But it's that which determines whether there should be a capital, not whether it is a verb or a noun.
If the verb is not deserving of a capital then it's corresponding noun should not either. If it is OK to talk about googling something, then the activity itself can be legitimately referred to as a 'google'.
Once you verb it, it becomes just another word like googling, photoshopped, videotaping, zippered, etc. However, some authors do ensure that they capitalise trademarks to avoid any infringement and/or mention that it's a trademark. Adobe, for e.g., can be quite vehement in their objection to the use of 'photoshop' as a verb.
Wiktionary has an entry for Skype. It also informs me that pesky is an anagram of Skype, which is quite ... appropriate.