Oy....sheesh. Kahane Loyalist is right. If you convert to Judaism you are as Jewish as a Jew who is born a Jew...and you do count yourself as descended from Avraham.
It is also true that over thousands of years...because of the Diaspora and the hundreds of different countries into which we were dispersed from Judah, the Ashkenazim into mainly European countries, The Sephardim into Spanish countries and the Mizrahi into Arabic ones....we Jews began to take on the characteristics of the nations we entered. This is due to intermarriage, and yes some of the native populations converting to Judaism and then marrying Jews. The kids are always going to have characteristics of their parents.
Now though! Now is the time when we are all coming back to Israel...we are a melting pot of all the different nations into which we fled. And NOW we have the chance to marry one another. Now, there is the opportunity of making one Israeli out of us all.