Hello,
I will start a site with a .asia domain name extension. I know that the domain name itself is of very big importance for a search result. The domain containing the keyword and closer to the keyword have great advantage. But I know this for .com domains only. Will this statement be true for .net .eu .asia .de .co.uk etc.... domains as well?

In other words, everything else being 100% same including the website and everything, would a .com domain name have more advantage in search engine results than the same domain with different extension?

Nope! the domain name is NOT of "big importance"

And you don't "know" this, you have read it "on the internet" and are assuming that "they" knew what they were talking about.

definitely yes... the domain name is of big importance. I did not read this anywhere at all... this statement is after a couple years of experience of building and running my own website(s), which all have keywords within the domain names. I have a few websites and they rank really well (in the first few results or a couple pages of google) for the words that they contain in their domain names. and i honestly never did any seo so far with any of them, but I am just updating them with fresh and original content frequently thats all...

Anyway, if we return to my question... is there any ideas?

And are any of these "keywords" remotely "competitive"?

none of them are very competitive, but they are not bad either... they provide me a very good google and yahoo ranking though.
After what you said, the only other possibility that I can think of is maybe the title tags being the same as or similar to the domain name, may be that is the reason.

EXACTLY!!!

The title element IS the thing that matters most of all.

Fyi, domain names have nothing to do with ranking. That's a fallacy. Google doesn't favor any specific domains and it doesn't matter whether or not your target keywords are included in your url. Ranking is based on link quality and popularity.

why would you target .asia?