SEO problems with h1

The problem is that the logo is tagged to h1 and when listing articles in blog layout, the category title is also h1.

Is it possible to disable the template logo and use custom html module in logo position?

Also it would be great if the template tagging the article titles as h1, when there is no category title and as h2 when there is category title visible.

It seems that the logo is tagged as h1 in other templates as well. The logo is not telling to search engines anything, so it should be customized.

you can disable the logo from the template manager, and then make your own menu position. there is several guides on how to do that here in the forum, thou most is for 1.5 and t3 framwork, it is mostlikely the same procedure..