Do comment flags deemed helpful count toward the Deputy and Marshall badge requirements, or is it only question and answer flags that count?
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Only the flags for moderator attention are counted for getting the Deputy, and Marshall badges; this, for example, excludes the flags for spam posts.
On a site, I didn't get the Marshall badge, even if my total helpful flags are more than 500; that is because my flags for moderator attention are just 387.
Update: Moderators don't get those badges, and the site I was referring to is the site where I am a moderator. It is not true that only flags for moderator attention are considered for getting those badges.
Yes they do. While can't find any official documentation for this, I can prove it with the help of Yannis who was kind enough to give screenshot of his flag history at http://workplace.stackexchange.com: (source)
And as you can see here, the badge is all his: