They are important, make your text(H1 H2 H3) larger then the text around, usually, the Search Engine will index your H1 content first. Also, If there's a paragraph before your H1, and the Bold Text is included in the paragraph, the Search Engine will more likely to index the Bold Text first. So i think, Bold keywords text is the most important.
the headsection is best suitable for h1 and h2 tags... it really gives nice view of the page too that gives highlight to the title of any topic or subject of the webpage.. crawler also consider them to give the ranking..
SEO is like cooking. Every little ingredient is important and your header tags are no different. It's one of many important things to do for on-page optimization.
Using the h tags can be extremely important for any website but they need to be used in the right way. They should be used sparingly and for the more important keywords/phrases that you are focusing on.
I would say they still have significant value if used effectively. I would use only one H1 tag per page.
Keep your keywords limited by 5 or 6 for best performance. Having more keywords in META tag doesn't work if those cannot impact your page content for density of each.
The keywords should consist in your body for more than 2% of total words between <body> and</body> tags. The keyword density upto 4 or 5% is better but more than 5% is considered as spamming.
Keywords are not to put between <body> and </body> tags. Keywords are META tags. And it is page content which are between <body> and </body> tags. Browser will display the contents between <body> and </body> tags as the output.