My site is over 2 months old but I see it's not ranked first place for my domain name when i search it. Is this usual?
Yup, This mean that you need to do better work, or that you hvae indexed your website in a high PR directory. When you submit a query about your website, Google bot will search for this directory 1st.
It depends on your domain name. If it contains a highly competetive phrase then it will indeed take time to rank for its name. If however, your domain is more like bigjohnsbigbluewidgetsonlineindenmark.com then there could be more to it. Have you got your site in Google Webmaster Tools - check to see how many pages are indexed and how many pages link to them. If you have few or no links then look to acquire some quickly. Forum posts will help, just make sure you do it legitimately rather than spamming.
You have more than 550 post only here in DP. place that link with anchor text here, And also place your link in profile, to i am damn sure, You will get good results soon. Best of Luck.
Two months is too much if you're doing the right thing, better check your SEO techniques if your doing such.
You should focus on your on screen and off screen optimization because there is some competition for your keyword.