Hi Rwaynick, yes I did that shortly after I started this post. It did not help me a lot, but at least system stopped going into the original URL. I am going to be closing this topic as resolved but I will be opening another one as I have several other problems starting with the missing wp-config.php that for me is just a softlink.
thanks for you help.
just a quick update, I noticed just now that when I try to go in http://localhost/mysite/wp-admin/ besides of WP changing the address to https… in tries to reach the IP where the original WP sites was hosted. What in WP is doing this?