Do heading tags really matter?

Selected answers from the Dumb SEO Questions Facebook & G+ community.

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  Jenny Halasz: Yes what you stated is expected behavior and compliant with HTML5 and W3c. The latter option isn`t necessarily a problem, but it`s not officially correct. Google only cares about the H1 and the first H2 at most.
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  Webado C Webada: There`s a hierarchy implied by the numbers. You should use that as a guideline of how to structure your content, and don`t use header tags for styling.
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  Neil Cheesman: Genesis Theme and their own Featured Posts which is part of the theme.... wiget-titles (ie title of the featured posts section) are H4 while the post titles that get shown are H2s - which to me doesn`t make sense.... I have emailed them but no satisfactory answer to date...
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  Neil Cheesman: I have tried some code re changing widget-titles from H4 to H2 but at the moment with no success...
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  Webado C Webada: But how did the h4 end up being used? Why?
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  Webado C Webada: Oh the header for featured posts stuff. Well that`s secondary on a page, so it`s OK.
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  Webado C Webada: The actual post or page titles should end up under h1`s. When the post is listed along with several others like on the homepage, each post`s title should be h2.
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  Webado C Webada: Meh, much depends on the options in the text editor. Some may be very poor depending on the theme. I hope that WP installation doesn`t set h1 to be the site name. It has to be the title of the page. The site name should be in a different world of its own.
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  Neil Cheesman: The reason I ask is that a Wordpress plugin re featured posts with their titles etc and it has the second option.. and it just doesn`t seem right... or make sense...
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  Webado C Webada: You`re overthinking it again lol
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  Neil Cheesman: okay.. but what if those post titles are listed under section headings... what should those section headings be? ie... California News H1 Today`s News H? It Rained H? (post title) Sport was played H? (post title) There was dancing in the street H? (post title) Today`s Sport H? Football H? post title Hockey H? post title Tennis H? post title
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  Webado C Webada: I can`t imagine what that h4 is doing. Again it must be according to the theme. What`s a plugin doing managing that anyway?
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  Neil Cheesman: Title and H1 are fine....
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  Webado C Webada: With html5 the ... is structured the same way as a page`s body. It can have its own h1, h2`s under that, etc.
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  Neil Cheesman: Not overthinking... that`s what there is.... section headings and post titles under them... does it matter what H tag they are or what would be suggested as preferable... or it doesn`t really matter one way or the other...