The H3 in the menu is for accessibility – it’s important, but not as important as h1 and h2s. Right now, there’s no way to change this, but I’ll look into adding a filter in a future version to make that possible.
The H1 for the page title is correct.
The widget titles are less important, so they’re all set as h4 tags, which is correct.
Then it’s up to you to use an H2 in your body content somewhere (preferably under the content title).
So basically, the only thing tripping up your validation is the menu H3, correct?
I cannot change the style, family, weight and so on of my headings (h1, h2, h3, h4, h5, h6) with style.css. If I change or add some font information in style.css nothings changes in my website.